☆ On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs

Luís Baptista, Matthias Hofmann

We study the structure of eigenfunctions of the Laplacian on quantum graphs, with a particular focus on Morse eigenfunctions via nodal and Neumann domains. Building on Courant-type arguments, we establish upper bounds for the number of nodal points and explore conditions under which Neumann domains of eigenfunctions correspond to minimizers to a class of spectral partition problems often known as spectral minimal partitions. The main focus will be the analysis on tree graphs, where we characterize the spectral energies of such partitions and relate them to the eigenvalues of the Laplacian under genericity assumptions. Notably, we introduce a notion analogous to Courant-sharpness for Neumann counts and demonstrate when spectral minimal partitions coincide with partitions formed by Neumann domains of eigenfunctions.

☆ Vorticity blow-up for the 3D incompressible Euler equations

Wenjie Deng, Song Jiang, Minling Li, Zhaonan Lou

In this paper, we study the finite-time blow-up for classical solutions of the 3D incompressible Euler equations with low-regularity initial vorticity. Applying the self-similar method and stability analysis of the self-similar system in critical Sobolev space, we prove that the vorticity of the axi-symmetric 3D Euler equations develops a finite-time singularity with certain scaling indices. Furthermore, we investigate the time integrability of the solutions. The proof is based on the new observations for the null structure of the transport term, and the parameter stability of the fundamental self-similar models.

☆ Global existence and decay of small solutions in a viscous half Klein-Gordon equation

Louis Garénaux, Björn de Rijk

We establish global existence and decay of solutions of a viscous half Klein-Gordon equation with a quadratic nonlinearity considering initial data, whose Fourier transform is small in L1 cap Linfty. Our analysis relies on the observation that nonresonant dispersive effects yield a transformation of the quadratic nonlinearity into a subcritical nonlocal quartic one, which can be controlled by the linear diffusive dynamics through a standard L1 - Linfty argument. This transformation can be realized by applying the normal form method of Shatah or, equivalently, through integration by parts in time in the associated Duhamel formula.

comment: 17 pages

☆ Perturbation theory of the compressible Navier-Stokes equations and its application

Kazuyuki Tsuda

In this article, a perturbation theory of the compressible Navier-Stokes equations in $\mathbb{R}^n$ $(n \geq 3)$ is studied to investigate decay estimate of solutions around a non-constant state. As a concrete problem, stability is considered for a perturbation system from a stationary solution $u_\omega$ belonging to the weak $L^n$ space. Decay rates of the perturbation including $L^\infty$ norm are obtained which coincide with those of the heat kernel. The proof is based on deriving suitable resolvent estimates with perturbation terms in the low frequency part having a parabolic spectral curve. Our method can be applicable to dispersive hyperbolic systems like wave equations with strong damping. Indeed, a parabolic type decay rate of a solution is obtained for a damped wave equation including variable coefficients which satisfy spatial decay conditions.

☆ Low-complexity approximations with least-squares formulation of the time-dependent Schr{ö}dinger equation

Mi-Song Dupuy, Virginie Ehrlacher, Clément Guillot

We propose new methods designed to numerically approximate the solution to the time dependent Schr{\"o}dinger equation, based on two types of ansatz: tensors, and approximation by a linear combination of gaussian wave packets. In both cases, the method can be seen as a restricted optimization problem, which can be solved by adapting either the Alternating Least Square algorithm in the tensor case, or some greedy algorithm in the gaussian wavepacket case. We also discuss the efficiency of both approaches.

☆ Mathematical Study of Reaction-Diffusion in Congested Crowd Motion

Noureddine Igbida, Fahd Karami, Driss Meskine

This paper establishes existence, uniqueness, and an L^1-comparison principle for weak solutions of a PDE system modeling phase transition reaction-diffusion in congested crowd motion. We consider a general reaction term and mixed homogeneous (Dirichlet and Neumann) boundary conditions. This model is applicable to various problems, including multi-species diffusion-segregation and pedestrian dynamics with congestion. Furthermore, our analysis of the reaction term yields sufficient conditions combining the drift with the reaction that guarantee the absence of congestion, reducing the dynamics to a constrained linear reaction-transport equation.

☆ Degenerate Elliptic PDEs on a Network with Kirchhoff Conditions

Guy Barles, Olivier Ley, Erwin Topp

In this article, we are interested in semilinear, possibly degenerate elliptic equations posed on a general network, with nonlinear Kirchhoff-type conditions for its interior vertices and Dirichlet boundary conditions for the boundary ones. The novelty here is the generality of the equations posed on each edge that is incident to a particular vertex, ranging from first-order equations to uniformly elliptic ones. Our main result is a strong comparison principle, i.e., a comparison result between discontinuous viscosity sub and supersolutions of such problems, from which we conclude the existence and uniqueness of a continuous viscosity by Perron's method. Further extensions are also discussed.

☆ Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy

Jean-Michel Coron, Joachim Krieger, Shengquan Xiang

We study wave maps from the circle to a general compact Riemannian manifold. We prove that the global controllability of this geometric equation is characterized precisely by the homotopy class of the data. As a remarkable intermediate result, we establish uniform-time global controllability between steady states, providing a partial answer to an open problem raised by Dehman, Lebeau and Zuazua (2003). Finally, we obtain quantitative exponential stability around closed geodesics with negative sectional curvature. This work highlights the rich interplay between partial differential equations, differential geometry, and control theory.

☆ Normalized solutions of quasilinear Schrödinger equation with Sobolev critical exponent on star-shaped bounded domains

Ru Yan

In this paper, we consider a quasilinear Schr\"odinger equation with critical exponent on bounded domains. Via a dual approach, we establish the existence of two positive normalized solutions: one is a ground state and the other is a mountain pass solution.

☆ Inverse scattering for the fractional Schrödinger equation

Saumyajit Das, Tuhin Ghosh, Shiqi Ma

This article is devoted to studying the inverse scattering for the fractional Schr\"{o}dinger equation, and in particular we solve the Born approximation problem. Based on the ($p$,$q$)-type resolvent estimate for the fractional Laplacian, we derive an expression for the scattering amplitude of the scattered solution of the fractional Schr\"{o}dinger equation. We prove the uniqueness of the potential using the scattering amplitude data.

comment: 19 pages

☆ Ill-posedness in $B^s_{p,\infty}$ of the Euler equations: Non-continuous dependence

Jinlu Li, Yanghai Yu

In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]{BCD}. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]{BCD} the existence and uniqueness of $B^s_{p,\infty}$ solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from $B^s_{p,\infty}$ to $L_T^\infty B^s_{p,\infty}$ for $s>1+d/p$ with $1\leq p\leq \infty$ and in the H\"{o}lder spaces from $C^{k,\alpha}$ to $L_T^\infty C^{k,\alpha}$ with $k\in \mathbb{N}^+$ and $\alpha\in(0,1)$, which later covers particularly the ill-posedness of $C^{1,\alpha}$ solution in \cite[Trans. Amer. Math. Soc., (2018)]{MYtams}. Beyond purely technical aspects on the choice of initial data, a remarkable novelty of the proof is the construction of an approximate solution to the Burgers equation.

☆ Global Existence and Incompressible Limit for the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data

Qinghao Lei

In this paper, we study the three-dimensional axisymmetric compressible Navier-Stokes equations with slip boundary conditions in a cylindrical domain excluding the axis. We establish the global existence and exponential decay of weak, strong, and classical solutions with large initial data and vacuum, under the assumption that the bulk viscosity coefficient is sufficiently large. Moreover, we demonstrate that as the bulk viscosity coefficient tends to infinity, the solutions of the compressible Navier-Stokes equations converge to those of the inhomogeneous incompressible Navier-Stokes equations.

comment: arXiv admin note: substantial text overlap with arXiv:2509.11260

♻ ☆ Delayed parabolic regularity for curve shortening flow

Arjun Sobnack, Peter M. Topping

Given two curves bounding a region of area $A$ that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time $A/\pi$. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an $L^1$ function.

comment: v1 was not submitted. v2 has numerous adjustments and improvements to the exposition

♻ ☆ A double-phase Neumann problem with $p=1$

Alexandros Matsoukas, Nikos Yannakakis

We study a double-phase Neumann problem with non-homogeneous boundary conditions, where the lowest exponent $p$ is equal to 1. The existence of a solution is established as the limit of solutions to corresponding double-phase problems with $p>1$. We also provide a variational characterization of the limit.

comment: 18 pages

♻ ☆ Uniqueness in determining multidimensional domains with unknown initial data

Jone Apraiz, Anna Doubova, Enrique Fernández-Cara, Masahiro Yamamoto

This paper addresses several geometric inverse problems for some linear parabolic systems where the initial data (and sometimes also the coefficients of the equations) are unknown. The goal is to identify a subdomain within a multidimensional set. The non-homogeneous part of the equation is expressed as a function satisfying some specific assumptions near a positive time. We establish uniqueness results by incorporating observations that can be on a part of the boundary or in an interior (small) domain. Through this process, we also derive information about the initial data. The main tools required for the proofs include semigroup theory, unique continuation and time analyticity results

♻ ☆ Upper bound on heat kernels of finite particle systems of Keller-Segel type

S. E. Boutiah, D. Kinzebulatov

We obtain an upper bound on the heat kernel of the Keller-Segel finite particle system that exhibits blow up effects. The proof exploits a connection between Keller-Segel finite particles and certain non-local operators. The latter allows to address some aspects of the critical behaviour of the Keller-Segel system resulting from its two-dimensionality.

comment: References added, a few typos fixed

♻ ☆ On the singular set of $\operatorname{BV}$ minimizers for non-autonomous functionals

Lukas Fußangel, Buddhika Priyasad, Paul Stephan

We investigate regularity properties of minimizers for non-autonomous convex variational integrands $F(x, \mathrm{D} u)$ with linear growth, defined on bounded Lipschitz domains $\Omega \subset \mathbb{R}^n$. Assuming appropriate ellipticity conditions and H\"older continuity of $\mathrm{D}_zF(x,z)$ with respect to the first variable, we establish higher integrability of the gradient of minimizers and provide bounds on the Hausdorff dimension of the singular set of minimizers.

comment: Version 2, 26 pages, 1 figure. Final version to appear in Communications in Contemporary Mathematics

♻ ☆ The Ground State of a Cubic-quintic Nonlinear Schrödinger Equation with Radial Potential in the Thomas-Fermi Limit

Deke Li, Qingxuan Wang

We focus on the ground state of the cubic-quintic nonlinear Schr\"{o}dinger energy functional \begin{gather*} \begin{aligned} {E}(\varphi)=\frac{1}{2}\int_{\mathbb{R}^d}\left(|\nabla \varphi|^2+V(x)|\varphi|^2\right)\,dx \pm\frac{1}{4}\int_{\mathbb{R}^d}|\varphi|^4\,dx +\frac{1}{6}\int_{\mathbb{R}^d}|\varphi|^6\,dx, (d=1,2,3) \end{aligned} \end{gather*} under the mass constraint $\int_{\mathbb{R}^d}|\varphi|^2\,dx=N$, where $N$ can be viewed as particle number, and $V(x)$ behaves like $C|x|^p (p\geq 2)$ as $|x|\rightarrow +\infty$, including the harmonic potential. When $N\rightarrow +\infty$, we show that up to a suitable scaling the ground state $\varphi_N$ would convergence strongly in some $L^q(\mathbb{R}^d)$ space to a Thomas-Fermi minimizer, this limit can be referred to as the \emph{Thomas-Fermi limit}. The limit Thomas-Fermi profile has compact support, given by $u^{TF}(x)=\left[\mu^{TF}-C_0|x|^p\right]^{\frac{1}{4}}_{+}$, where $\mu^{TF}$ is a suitable Lagrange multiplier with exact value. We find that, similar to the asymptotic analysis in [J. Funct. Anal. 260 (2011), 2387-2406.] and [Arch. Ration. Mech. Anal. 217 (2015), 439-523.] for Gross-Pitaevskii energy in the Thomas-Fermi limit where a small parameter $\varepsilon$ tends to 0, there also has a steep \emph{corner layer} near the boundary of compact support of $u^{TF}(x)$, in which the ground state has irregular behavior as $N\rightarrow +\infty$. Finally, we establish a new energy method to obtain the $L^\infty$-convergence rates of ground states $\varphi_N$ inside the corner layer and outside corner layer respectively, this method may be applicable to other general nonlinearities.

comment: 39 pages

♻ ☆ Construction of solutions for a critical elliptic system of Hamiltonian type

Yuxia Guo, Congzheng Xuanyuan, Tingfeng Yuan

We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*}\left\{\begin{aligned} &-\Delta u + V(|y'|,y'') u = v^p, \;\; \text{in} \;\; \mathbb{R}^N,\\ &-\Delta v + V(|y'|,y'') v = u^q, \;\; \text{in} \;\; \mathbb{R}^N,\\ &u, v > 0 , (u,v) \in (\dot{W}^{2,\frac{p+1}{p}} (\mathbb{R}^N) \cap L^2(\mathbb{R}^N)) \times (\dot{W}^{2,\frac{q+1}{q}} (\mathbb{R}^N) \cap L^2(\mathbb{R}^N)) ,\end{aligned}\right. \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$ and $V(|y'|, y'')\not\equiv 0$ is a bounded non-negative function in $\mathbb{R}_+\times \mathbb{R}^{N-2}$, $p,q>1$ satisfying $$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$$ By using a finite reduction argument and local Pohozaev identities, under the assumption that $N\geq 5$, $(p,q)$ lies in the certain range and $r^2V(r,y'')$ has a stable critical point, we prove that the above problem has infinitely many solutions whose energy can be made arbitrarily large.

♻ ☆ Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities

Yi-Hsuan Lin, Teemu Tyni, Philipp Zimmermann

The main purpose of this article is to establish the Runge-type approximation in $L^2(0,T;\widetilde{H}^s(\Omega))$ for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calder\'on problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calder\'on problem of nonlocal wave equations with polyhomogeneous nonlinearities.

comment: 38 pages

☆ On the Fixed Point Property in Reflexive Banach Spaces

Faruk Alpay, Hamdi Alakkad

Fixed point theory studies conditions under which nonexpansive maps on Banach spaces have fixed points. This paper examines the open question of whether every reflexive Banach space has the fixed point property. After surveying classical results, we propose a quantitative framework based on diametral l1 pressure and weighted selection functionals, which measure how much an orbit hull of a fixed point free nonexpansive map can collapse. We prove that if either invariant is uniformly positive, then the space must contain a copy of l1 and thus cannot be reflexive. We present finite dimensional certificates, positive and negative examples, and an x86-64 routine that computes mutual coherence and a lower bound for the pressure. The paper clarifies why existing approaches fail and outlines open problems and ethical considerations.

comment: 35 pages, 1 fig., asm

☆ A study on state spaces in classical Banach spaces

Soumitra Daptari, Saurabh Dwivedi

Let $X$ be a real or complex Banach space. Let $S(X)$ denote the unit sphere of $X$. For $x\in S(X)$, let $S_{x}=\{x^*\in S(X^*):x^*(x)=1\}$. A lot of Banach space geometry can be determined by the `quantum' of the state space $S_{x}$. In this paper, we mainly study the norm compactness and weak compactness of the state space in the space of Bochner integrable function and $c_{0}$-direct sums of Banach spaces. Suppose $X$ is such that $X^*$ is separable and let $\mu$ be the Lebesgue measure on $[0,1]$. For $f\in L^1(\mu,X)$, we demonstrate that if $S_{f}$ is norm compact, then $f$ is a smooth point. When $\mu$ is the discrete measure, we show that if $ (x_i) \in S(\ell^{1}(X))$ and $ \|x_{i}\|\neq 0$ for all $i\in{\mathbb{N}}$, then $ S_{(x_i)}$ is weakly compact in $ \ell^\infty(X^*) $ if and only if $ S_{\frac{x_i}{\|x_i\|}} $ is weakly compact in $X^*$ for each $i\in{\mathbb{N}}$ and $\text{diam}\left(S_{\frac{x_i}{\|x_i\|}}\right) \to 0 $. For discrete $c_{0}$-sums, we show that for $(x_{i})\in c_{0}(X)$, $S_{(x_{i})}$ is weakly compact if and only if for each $i_{0}\in \mathbb{N}$ such that $\|x_{i_{0}}\|=1$, the state space $S_{x_{i_{0}}}$ is weakly compact.

comment: To appear in Colloquium Mathematicum

♻ ☆ On p-summability in weighted Banach spaces of holomorphic functions

M. G. Cabrera-Padilla, A. Jiménez-Vargas, A. Keten Çopur

Given an open subset $U$ of a complex Banach space $E$, a weight $v$ on $U$, and a complex Banach space $F$, let $\Hv(U,F)$ denote the Banach space of all weighted holomorphic mappings $f\colon U\to F$, under the weighted supremum norm $\left\|f\right\|_v:=\sup\left\{v(x)\left\|f(x)\right\|\colon x\in U\right\}$. In this paper, we introduce and study the class $\Pi_p^{\Hv}(U,F)$ of $p$-summing weighted holomorphic mappings. We prove that it is an injective Banach ideal of weighted holomorphic mappings. Variants for weighted holomorphic mappings of Pietsch Domination Theorem, Pietsch Factorization Theorem and Maurey Extrapolation Theorem are presented. We also identify the spaces of $p$-summing weighted holomorphic mappings from $U$ into $F^*$ under the norm $\pi^{\Hv}_p$ with the duals of $F$-valued $\Hv$-molecules on $U$ under a suitable version $d^{\Hv}_{p^*}$ of the Chevet--Saphar tensor norms.

comment: 25 pages

♻ ☆ Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions

Konstantinos Bampouras, Karl-Mikael Perfekt

We consider Schatten class membership of Hankel operators on Paley--Wiener spaces of convex $\Omega \subset \mathbb{R}^n$, both for bounded and unbounded domains. In particular, the classical product Hardy spaces fit within our theory. For admissible domains, we develop a framework and theory of Besov spaces of Paley--Wiener type, and prove that a Hankel operator belongs to the Schatten class $S^p$ if and only if its symbol belongs to a corresponding Besov space, for $1 \leq p \leq 2$. We extend this result to all $1 \leq p < \infty$ for the classical product Hardy spaces and for the Paley--Wiener space of a bounded smooth domain $\Omega \subset \mathbb{R}^n$ of strictly positive curvature.

comment: This article supersedes a previous preprint of the authors. The main results now characterize the Schatten class Hankel operators on the product Hardy space as well as for the Paley Wiener space associated with any bounded strongly convex set in Rn. The results apply for all parameters of the weights, and for all p between 1 and infinity. More general results are also presented

♻ ☆ Parallel methods for quasinonexpansive mappings in a Hilbert space

Koji Aoyama, Shigeru Iemoto

This paper is devoted to the problem of finding a common fixed point of quasinonexpansive mappings defined on a Hilbert space. To approximate the solution to this problem, we present several iterative processes using the parallel method based on Anh and Chung (2014) and Aoyama (2018).

☆ Dynamic Decision Modeling for Viable Short and Long Term Production Policies: An HJB Approach

Achraf Bouhmady, Mustapha Serhani, Nadia Raissi

This study introduces a mathematical framework to investigate the viability and reachability of production systems under constraints. We develop a model that incorporates key decision variables, such as pricing policy, quality investment, and advertising, to analyze short-term tactical decisions and long-term strategic outcomes. In the short term, we constructed a capture basin that defined the initial conditions under which production viability constraints were satisfied within the target zone. In the long term, we explore the dynamics of product quality and market demand to achieve and sustain the desired target. The Hamilton-Jacobi-Bellman (HJB) theory characterizes the capture basin and viability kernel using viscosity solutions of the HJB equation. This approach, which avoids controllability assumptions, is well suited to viability problems with specified targets. It provides managers with insights into maintaining production and inventory levels within viable ranges while considering product quality and evolving market demand. We numerically studied the HJB equation to design and test computational methods that validate the theoretical insights. Simulations offer practical tools for decision-makers to address operational challenges while aligning with the long-term sustainability goals. This study enhances the production system performance and resilience by linking rigorous mathematics with actionable solutions.

☆ Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations

Alekos Cecchin, Alessandro Goffi

We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton-Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form $\partial_t u+H(x,t,Du,D^2u)=0$ in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator $\varepsilon F(x,t,D^2u)$. Assuming that $u\in C^\alpha_x$, $u_0\in C^\eta$, $H\in C^\beta_x$ and having power growth $\gamma$ in the gradient entry, we establish a convergence rate of order $\varepsilon^{\min\left\{\frac{\eta}{2},\frac{\beta+\gamma(\alpha-1)}{\beta+\gamma(\alpha-1)+2-\alpha}\right\}}$. Our novel approach exploits the regularizing properties of sup/inf-convolutions for viscosity solutions and the comparison principle. We also obtain explicit constants and do not assume differentiability properties neither on solutions nor on $H$. The same method provides new convergence rates for the vanishing viscosity approximation of the stationary counterpart of the equation and for transport equations with H\"older coefficients.

☆ Multidimensional Scalar Conservation Laws with Non-Aligned Discontinuous Flux and Singularity of Solutions

Ajlan Zajmović

We prove that the family of solutions to vanishing viscosity approximation for multidimensional scalar conservation laws with discontinuous non-aligned flux and zero initial data in the limit generates a singular measure supported along the discontinuity surface.

comment: 23 pages, 1 figure

☆ The Polya-Szego principle in the fractional setting: a glimpse on nonlocal functional inequalities

Alessandro Carbotti

In this survey we present the fractional Polya Szego principle and its main consequences in the study of nonlocal functional inequalities. In particular, we show how symmetrization methods work also in the fractional setting and yield sharp results such as isoperimetric type inequalities. Further developments including stability issues and generalizations in the anisotropic and the Gaussian setting are also discussed.

comment: 22 pages, 1 figure

☆ X-ray imaging from nonlinear waves: numerical reconstruction of a cubic nonlinearity

Markus Harju, Suvi Takalahti, Teemu Tyni

We study an inverse boundary value problem for the nonlinear wave equation in $2 + 1$ dimensions. The objective is to recover an unknown potential $q(x, t)$ from the associated Dirichlet-to-Neumann map using real-valued waves. We propose a direct numerical reconstruction method for the Radon transform of $q$, which can then be inverted using standard X-ray tomography techniques to determine $q$. Our implementation introduces a spectral regularization procedure to stabilize the numerical differentiation step required in the reconstruction, improving robustness with respect to noise in the boundary data. We also give rigorous justification and stability estimates for the regularized spectral differentiation of noisy measurements. A direct pointwise reconstruction method for $q$ is also implemented for comparison. Numerical experiments demonstrate the feasibility of recovering potentials from boundary measurements of nonlinear waves and illustrate the advantages of the Radon-based reconstruction.

comment: 22 pages, 8 figures

☆ Entire Large Solutions for Competitive Semilinear Elliptic Systems with General Nonlinearities Satisfying Keller--Osserman Conditions

Dragos-Patru Covei

We generalize a theorem of Lair concerning the existence of positive entire large solutions to competitive semilinear elliptic systems. While Lair's original result \cite{Lair2025} was established for power-type nonlinearities, our work extends the theory to a broad class of general nonlinearities satisfying a Keller--Osserman-type growth condition. The proof follows the same conceptual framework monotone iteration to construct global positive solutions, reduction to a scalar inequality for the sum of the components, application of a Keller--Osserman transform, and a two-step radial integration argument but replaces the explicit power-law growth with a general monotone envelope function. This approach yields a unified and verifiable criterion for the existence of large solutions in terms of the Keller--Osserman integral, thereby encompassing both critical and supercritical growth regimes within a single analytical setting.

comment: 12 pages

☆ A thermodynamically consistent model for bulk-surface viscous fluid mixtures: Model derivation and mathematical analysis

Patrik Knopf, Jonas Stange

We derive and analyze a new diffuse interface model for incompressible, viscous fluid mixtures with bulk-surface interaction. Our system consists of a Navier--Stokes--Cahn--Hilliard model in the bulk that is coupled to a surface Navier--Stokes--Cahn--Hilliard model on the boundary. Compared with previous models, the inclusion of an additional surface Navier--Stokes equation is motivated, for example, by biological applications such as the seminal \textit{fluid mosaic model} (Singer \& Nicolson, \textit{Science}, 1972) in which the surface of biological cells is interpreted as a thin layer of viscous fluids. We derive our new model by means of local mass balance laws, local energy dissipation laws, and the Lagrange multiplier approach. Moreover, we prove the existence of global weak solutions via a semi-Galerkin discretization. The core part of the mathematical analysis is the study of a novel bulk-surface Stokes system and its corresponding bulk-surface Stokes operator. Its eigenfunctions are used as the Galerkin basis to discretize the bulk-surface Navier--Stokes subsystem.

☆ A nonlinear model for long-range segregation

Howen Chuah, Stefania Patrizi, Monica Torres

We study a system of fully nonlinear elliptic equations, depending on a small parameter $\eps$, that models long-range segregation of populations. The diffusion is governed by the negative Pucci operator. In the linear case, this system was previously investigated by Caffarelli, the second author, and Quitalo in \cite{CL2} as a model in population dynamics. We establish the existence of solutions and prove convergence as $\eps\to0^+$ to a free boundary problem in which populations remain segregated at a positive distance. In addition, we show that the supports of the limiting functions are sets of finite perimeter and satisfy a semi-convexity property.

comment: 23 pages

☆ A survey on anisotropic integral representation results

Simone Verzellesi

In this note we review some recent results concerning integral representation properties of local functionals driven by Lipschitz continuous anisotropies.

comment: This note is a review, and it does not contain new results. The reader is referred to the original sources according to the note's suggestions

☆ A note on Tricomi-type partial differential equations with white noise initial condition

Enrico Bernardi, Alberto Lanconelli

We study a class of Tricomi-type partial differential equations previously investigated in [28]. Firstly, we generalize the representation formula for the solution obtained there by allowing the coefficient in front of the second-order partial derivative with respect to $x$ to be any non integer power of $t$. Then, we analyze the robustness of that solution by taking the initial data to be Gaussian white noise and we discover that the existence of a well-defined random field solution is lost upon the introduction of lower-order terms in the operator. This phenomenon shows that, even though the Tricomi-type operators with or without lower-order terms are the same from the point of view of the theory of hyperbolic operators with double characteristics, their corresponding random versions exhibit different well posedness properties. We also prove that for more regular initial data, specifically fractional Gaussian white noise with Hurst parameter $H\in (1/2,1)$, the well posedness of the Cauchy problem for the Tricomi-type operator with lower-order term is restored.

comment: 14 pages

☆ A Survey on the Div-Curl Lemma and Some Extensions to Fractional Sobolev Spaces

Maicol Caponi

This survey is a chapter of a forthcoming book. This chapter recalls the classical formulation of the Div-Curl lemma along with its proof, and presents some possible generalizations in the fractional setting, within the framework of the Riesz fractional gradient and divergence introduced by Shieh and Spector (2015) and further developed by Comi and Stefani (2019).

☆ Lie symmetry analysis and similarity reductions for the tempered-fractional Keller Segel system

Ghorbanali Haghighatdoost, Mustafa Bazghandi

We perform a Lie symmetry analysis on the tempered-fractional Keller Segel (TFKS) system, a chemo-taxis model incorporating anomalous diffusion. A novel approach is used to handle the nonlocal nature of tempered fractional operators. By deriving the full set of Lie point symmetries and identifying the optimal one-dimensional subalgebras, we reduce the TFKS PDEs to ordinary differential equations (ODEs), yielding new exact solutions. These results offer insights into the long-term behavior and aggregation dynamics of the TFKS model and present a methodology applicable to other tempered fractional differential equations.

comment: 11 pages

☆ Random data Cauchy theory for fully nonlocal telegraph equations

Xi Huang, Li Peng, Juan Carlos Pozo, Yong Zhou

We consider the random Cauchy problem for the fully nonlocal telegraph equation of power type with the general $(\mathcal{PC}^{\ast})$ type kernel $(a,b)$. This equation can effectively characterize high-frequency signal transmission in small-scale systems. We establish a new completely positive kernel induced by $b$ (see Appendix \refeq{app b}) and derive two novel solution operators by using the relaxation functions associated with the new kernel,which are closely related to the operators $\cos(\theta(-\Delta)^{\frac{\beta}{4}} )$ and $(-\Delta)^{-\frac{\beta}{4} }\sin(\theta(-\Delta)^{\frac{\beta}{4}} )$ for $\beta\in(1,2]$. These operators enable, for the first time, the derivation of mixed-norm $L_t^qL_x^{p'}$ estimates for the novel solution operators. Next, utilizing probabilistic randomization methods, we establish the average effects, the local existence and uniqueness for a large set of initial data $u^\omega \in L^{2}(\Omega, H^{s,p}(\mathbb R^3))$ ($p\in (1,2)$) while also obtaining probabilistic estimates for local existence under randomized initial conditions. The results reveal a critical phenomenon in the temporal regularity of the solution regarding the regularity index $s$ of the initial data $u^\omega$.

☆ On the logarithmic correction of transition fronts in shifting environments

King-Yeung Lam, Chang-Hong Wu

In this paper, we investigate the location of the spreading front and convergence to traveling wave profile of solutions to the Fisher-KPP equation in the following two cases: (i) in unbounded domains with an expanding boundary; (ii) on the real line where the environment function has a shifting jump discontinuity. Our approach is based on extending ideas in Bramson's seminal work in 1983, and applying gluing technique to construct super/subsolutions.

☆ Solvability of some integro-differential equations with the bi-Laplacian and transport

Vitali Vougalter

We demonstrate the existence in the sense of sequences of solutions for some integro-differential type problems involving the drift term and the square of the Laplace operator, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding H^4 spaces. Our argument is based on the fixed point technique when the elliptic equations contain fourth order differential operators with and without the Fredholm property. It is established that, under the reasonable technical conditions, the convergence in L^1 of the integral kernels yields the existence and convergence in H^4 of the solutions.

☆ Convergence of a Second-Order Projection Method to Leray-Hopf Solutions of the Incompressible Navier-Stokes Equations

Franziska Weber

We analyze a second-order projection method for the incompressible Navier-Stokes equations on bounded Lipschitz domains. The scheme employs a Backward Differentiation Formula of order two (BDF2) for the time discretization, combined with conforming finite elements in space. Projection methods are widely used to enforce incompressibility, yet rigorous convergence results for possibly non-smooth solutions have so far been restricted to first-order schemes. We establish, for the first time, convergence (up to subsequence) of a second-order projection method to Leray-Hopf weak solutions under minimal assumptions on the data, namely $u_0 \in L^2_{\text{div}}(\Omega)$ and $f \in L^2(0,T;L^2_{\text{div}}(\Omega))$. Our analysis relies on two ingredients: A discrete energy inequality providing uniform $L^{\infty}(0,T;L^2(\Omega))$ and $L^2(0,T;H^1_0(\Omega))$ bounds for suitable interpolants of the discrete velocities, and a compactness argument combining Simon's theorem with refined time-continuity estimates. These tools overcome the difficulty that only the projected velocity satisfies an approximate divergence-free condition, while the intermediate velocity is controlled in space. We conclude that a subsequence of the approximations converges to a Leray-Hopf weak solution. This result provides the first rigorous convergence proof for a higher-order projection method under no additional assumptions on the solution beyond those following from the standard a priori energy estimate.

☆ Effects of temporal variations on wave speeds of bistable traveling waves for Lotka-Volterra competition systems

Weiwei Ding, Zhanghua Liang

This paper investigates the bistable traveling waves for two-species Lotka-Volterra competition systems in time periodic environments. We focus especially on the influence of the temporal period, with existence results established for both small and large periods.We also show the existence of, and derive explicit formulas for, the limiting speeds as the period tends to zero or infinity, and provide estimates for the corresponding rates of convergence. Furthermore, we analyze the sign of wave speed. Assuming that both species share identical diffusion rates and intraspecific competition rates, we obtain a criterion for determining the sign of wave speed by comparing the intrinsic growth rates and interspecific competition strengths. More intriguingly, based on our explicit formulas for the limiting speeds, we construct an example in which the sign of wave speed changes with the temporal period. This example reveals that temporal variations can significantly influence competition outcomes,enabling different species to become dominant under different periods.

☆ Nonlinear stability of the Larson-Penston collapse

Yan Guo, Mahir Hadzic, Juhi Jang, Matthew Schrecker

We prove nonlinear stability of the Larson-Penston family of self-similarly collapsing solutions to the isothermal Euler-Poisson system. Our result applies to radially symmetric perturbations and it is the first full nonlinear stability result for radially imploding compressible flows. At the heart of the proof is the ground state character of the Larson-Penston solution, which exhibits important global monotonicity properties used throughout the proof. One of the key challenges is the proof of mode-stability for the non self-adjoint spectral problem which arises when linearising the dynamics around the Larson-Penston collapsing solution. To exclude the presence of complex growing modes other than the trivial one associated with time translation symmetry, we use a high-order energy method in low and high frequency regimes, for which the monotonicity properties are crucially exploited, and use rigorous computer-assisted techniques in the intermediate regime. In addition, the maximal dissipativity of the linearised operator is proven on arbitrary large backward light cones emanating from the singular point using the global monotonicity of the Larson-Penston solutions. Such a flexibility in linear analysis also facilitates nonlinear analysis and allows us to identify the exact number of derivatives necessary for the nonlinear stability statement. The proof is based on a two-tier high-order weighted energy method which ties bounds derived from the Duhamel formula to quasilinear top order estimates. To prove global existence we further use the Brouwer fixed point theorem to identify the final collapse time, which suppresses the trivial instability caused by the time-translation symmetry of the system.

comment: 149 pages, 1 figure

☆ On the absence of anomalous dissipation for the Navier-Stokes equations with Navier boundary conditions: a sufficient condition

Claude Bardos, Daniel W. Boutros, Edriss S. Titi

We consider the three-dimensional incompressible Navier-Stokes equations in a bounded domain with Navier boundary conditions. We provide a sufficient condition for the absence of anomalous energy dissipation without making assumptions on the behaviour of the corresponding pressure near the boundary or the existence of a strong solution to the incompressible Euler equations with the same initial data. We establish our result by using our recent regularity results for the pressure corresponding to weak solutions of the incompressible Euler equations [Arch. Ration. Mech. Anal., 249 (2025), 28].

comment: 14 pages

☆ Quantitative Scattering for the Energy-Critical Wave Equation on Asymptotically Flat Spacetimes

Benjamin Dodson, Shi-Zhuo Looi

The scattering theory for the energy-critical wave equation on asymptotically flat spacetimes has, to date, been qualitative. While the qualitative scattering of solutions is well-understood, explicit bounds on the solution's global spacetime norms have been unavailable in this geometric setting. This paper establishes an explicit, exponential-type global bound on the Strichartz norm $\| u\|_{L^8_{t,x}}$ for solutions to the defocusing equation $\Box_g u=u^5$, where $\Box_g$ is the d'Alembertian associated with the perturbed metric. The bound depends on the solution's energy and an \textit{a priori} $\dot H^5 \times \dot H^4$ regularity bound on the solution. The proof develops a strategy that bypasses the need for vector-field commutators. It combines an interaction Morawetz estimate adapted to variable coefficients to control the solution's recent history with a dispersive analysis founded on integrated local energy decay to control the remote past. This strategy, in turn, necessitates the regularity and specific decay assumptions on the metric. As a result, this work upgrades the existing qualitative scattering theory to a fully quantitative statement, which provides a concrete measure of the global behavior of solutions in this geometric setting.

comment: 22 pages

☆ Multiscale modelling, analysis and simulation of cancer invasion mediated by bound and soluble enzymes

Mariya Ptashnyk, Chandrasekhar Venkataraman

We formulate a cell-scale model for the degradation of the extra-cellular matrix by membrane-bound and soluble matrix degrading enzymes produced by cancer cells. Based on the microscopic model and using tools from the theory of homogenisation we propose a macroscopic model for cancer cell invasion into the extra-cellular matrix mediated by bound and soluble matrix degrading enzymes. For suitable and biologically relevant initial data we prove the macroscopic model is well-posed. We propose a finite element method for the numerical approximation of the macroscopic model and report on simulation results illustrating the role of the bound and soluble enzymes in cancer invasion processes.

☆ Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces

Tommaso Bruno, Effie Papageorgiou

Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $\Delta$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $\lambda_{0}$, and $\sigma \in (0,1)$. The aim of this paper is twofold. On the one hand, we determine the Fujita exponent for the fractional heat equation \[\partial_{t} u + \Delta^{\sigma}u = e^{\beta t}|u|^{\gamma-1}u,\] by proving that nontrivial positive global solutions exist if and only if $\gamma\geq 1 + \beta/ \lambda_{0}^{\sigma}$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation \[ \Delta^{\sigma} v - \lambda^{\sigma} v - v^{\gamma}=0 \] for $0\leq \lambda \leq \lambda_{0}$ and $1<\gamma< \frac{n+2\sigma}{n-2\sigma}$. The two problems are known to be connected and the latter, aside from its independent interest, is actually instrumental to the former. \smallskip At the core of our results stands a novel fractional Poincar\'e-type inequality expressed in terms of a new scale of $L^{2}$ fractional Sobolev spaces, which sharpens those known so far, and which holds more generally on Riemannian symmetric spaces of non-compact type. We also establish an associated Rellich--Kondrachov-like compact embedding theorem for radial functions, along with other related properties.

♻ ☆ Ergodicity and mixing for locally monotone stochastic evolution equations

Gerardo Barrera, Jonas M. Tölle

We establish general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated exponentially ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the $e$-property of the semigroup. Furthermore, we provide quantitative upper bounds for the Wasserstein $\varepsilon$-mixing times. Examples on possibly unbounded domains include the stochastic incompressible 2D Navier-Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic heat equation, as well as, stochastic semilinear equations such as the 1D stochastic Burgers equation.

comment: 44 pages, 98 references; major revision

♻ ☆ Short-time existence of Lagrangian mean curvature flow

Spandan Ghosh

In his paper `Conjectures on Bridgeland Stability', Joyce asked if one can desingularise the transverse intersection point of an immersed Lagrangian using JLT expanders such that one gets a Lagrangian mean curvature flow via the desingularisations. Begley and Moore answered this in the affirmative by constructing a family of desingularisations and showing that a certain limit along their flows satisfies LMCF along with convergence to the immersed Lagrangian in the sense of varifolds. We prove that there exists a solution with convergence in a stronger sense, using the notion of manifolds with corners and a-corners as introduced by Joyce. Our methods are a direct P.D.E. based approach, along the lines of the proof of short-time existence for network flow by Lira, Mazzeo, Pluda and Saez.

comment: 61 pages, 3 figures. (v2) several typos corrected, added figures

♻ ☆ Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation

Guopeng Li, Liying Tao, Tengfei Zhao

We consider the Cauchy problem for the stochastic Hartree nonlinear wave equations (SHNLW) with a cubic convolution nonlinearity and an additive stochastic forcing on the Euclidean space. Our goal in this paper is two-fold. (i) We study the defocusing energy-critical SHNLW on $\mathbb{R}^d$, for $d \geq 5$, and prove that they are globally well-posed with deterministic initial data in the energy space. (ii) Next, we consider the well-posedness of the defocusing energy-critical SHNLW with randomized initial data below the energy space. In particular, when $d=5$, we prove it is almost surely globally well-posed. As a byproduct, by removing the stochastic forcing our result covers the study of the (deterministic) Hartree nonlinear wave equation (HNLW) with randomized initial data below the energy space. The main ingredients in the globalization argument involve the probabilistic perturbation approach by B\'enyi-Oh-Pocovnicu (2015) and Pocovnicu (2017), time integration by parts trick of Oh-Pocovnicu (2016), and an estimate of the Hartree potential energy.

comment: 32 pages. To appear in Stochastics and Partial Differential Equations: Analysis and Computations

♻ ☆ Robust and fast iterative method for the elliptic Monge-Ampère equation

R. N. Köhle, K. T. W. Menting, K. Mitra, J. H. M. ten Thije Boonkkamp

This paper introduces a fast and robust iterative scheme for the elliptic Monge-Amp\`ere equation with Dirichlet boundary conditions. The Monge-Amp\`ere equation is a nonlinear and degenerate equation, with applications in optimal transport, geometric optics, and differential geometry. The proposed method linearises the equation and uses a fixed-point iteration (L-scheme), solving a Poisson problem in each step with a weighted residual as the right-hand side. This algorithm is robust against discretisation, nonlinearities, and degeneracies. For a weight greater than the largest eigenvalue of the Hessian, contraction in $H^2$ and $L^\infty$ is proven for both classical and generalised solutions, respectively. The method's performance can be enhanced by using preconditioners or Green's functions. Test cases demonstrate that the scheme outperforms Newton's method in speed and stability.

comment: 27 pages, 8 figures

♻ ☆ Rate-form equilibrium for an isotropic Cauchy-elastic formulation: Part I: modeling

Patrizio Neff, Nina J. Husemann, Sebastian Holthausen, Franz Gmeineder, Thomas Blesgen

We derive the rate-form spatial equilibrium system for a nonlinear Cauchy elastic formulation in isotropic finite-strain elasticity. For a given explicit Cauchy stress-strain constitutive equation, we determine those properties that pertain to the appearing fourth-order stiffness tensor. Notably, we show that this stiffness tensor $\mathbb{H}^{\text{ZJ}}(\sigma)$ acting on the Zaremba-Jaumann stress rate is uniformly positive definite. We suggest a mathematical treatment of the ensuing spatial PDE-system which may ultimately lead to a local existence result, to be presented in part II of this work. As a preparatory step, we show existence and uniqueness of a subproblem based on Korn's first inequality and the positive definiteness of this stiffness tensor. The procedure is not confined to Cauchy elasticity, however in the Cauchy elastic case, most theoretical statements can be made explicit. Our development suggests that looking at the rate-form equations of given Cauchy-elastic models may provide additional insight to the modeling of nonlinear isotropic elasticity. This especially concerns constitutive requirements emanating from the rate-formulation, here being reflected by the positive definiteness of $\mathbb{H}^{\text{ZJ}}(\sigma)$.

♻ ☆ Lower semicontinuity, Stoilow factorization and principal maps

Kari Astala, Daniel Faraco, André Guerra, Aleksis Koski, Jan Kristensen

We consider a strengthening of the usual quasiconvexity condition of Morrey in two dimensions, which allows us to prove lower semicontinuity for functionals which are unbounded as the determinant vanishes. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and \v{S}ver\'ak, plays a prominent role in our approach.

comment: 34 pages

♻ ☆ Quantitative Propagation of Chaos for 2D Viscous Vortex Model on the Whole Space

Xuanrui Feng, Zhenfu Wang

We derive the quantitative estimates of propagation of chaos for the large interacting particle systems in terms of the relative entropy between the joint law of the particles and the tensorized law of the mean field PDE. We resolve this problem for the first time for the viscous vortex model that approximates 2D Navier-Stokes equation in the vorticity formulation on the whole space. We obtain as key tools the Li-Yau-type estimates and Hamilton-type heat kernel estimates for 2D Navier-Stokes in the whole space.

comment: 26 pages

♻ ☆ An epsilon-regularity result for Griffith almost-minimizers in the plane

Manuel Friedrich, Camille Labourie, Kerrek Stinson

We present regularity results for the crack set of a minimizer for the Griffith fracture energy, arising in the variational modeling of brittle materials. In the planar setting, we prove an epsilon-regularity theorem showing that the crack is locally a $C^{1,1/2}$ curve outside of a singular set of zero Hausdorff measure. The main novelty is that, in contrast to previous results, no topological constraints on the crack are required. The results also apply to almost-minimizers.

comment: Separation of prior version

♻ ☆ Strichartz and local smoothing estimates for the fractional Schrödinger equations over fractal time

Jin Bong Lee, Sanghyuk Lee, Luz Roncal

We obtain Strichartz-type estimates for the fractional Schr\"odinger operator $f \mapsto e^{it(-\Delta)^{\gamma/2}} f$ over a time set $E$ of fractal dimension. To obtain those estimates capturing fractal nature of $E$, we employ the notions in the spirit of the Assouad dimension, such as, bounded Assouad characteristic and Assouad specturm. We also prove the estimate $$ \| e^{it(-\Delta)^{\gamma/2}} f \|_{L_t^q(\mathrm{d}\mu; L_x^r(\mathbb{R}^d))} \le C \|f\|_{H^s}, $$ where $\mu$ is a measure satisfying an $\alpha$-dimensional growth condition. In addition, we establish related inhomogeneous estimates and $L^2$ local smoothing estimates. A surprising feature of our work is that, despite dealing with rough fractal sets, we extend the known estimates for the fractional Schr\"odinger operators in a natural way, precisely consistent with the associated fractal dimensions.

♻ ☆ An existence and uniqueness result to evolution equations with sign-changing pseudo-differential operators and its applications to logarithmic Laplacian operators and second-order differential operators without ellipticity

Jae-Hwan Choi, Ildoo Kim

We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator $\log (-\Delta)$ and a second-order differential operator whose leading coefficients are not positive semi-definite.

comment: 49 pages; The paper has been accepted for publication in "Journal of Pseudo-Differential Operators and Applications"

♻ ☆ The new observations about the parameter-dependent Schrödinger-Poisson system

Chen Huang, Sihua Liang, Lei Ma

In this paper, we study the existence results of solutions for the following Schr\"{o}dinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -\Delta u+V(x)u-\lambda \phi u=f(u)&\quad\text{in}~\mathbb R^3, -\Delta\phi=u^2&\quad\text{in}~\mathbb R^3. \end{cases} \end{equation*} We first consider the case that the potential $V$ is positive and radial so that the mountain pass theorem could be implied. The other case is that the potential $V$ is coercive and sign-changing, which means that the Schr\"{o}dinger operator $-\Delta +V$ is allowed to be indefinite. To deal with this more difficult case, by a local linking argument and Morse theory, the system has a nontrivial solution. Furthermore, we also show the asymptotical behavior result of this solution. Additionally, the proofs rely on new observations regarding the solutions of the Poisson equation. As a main novelty with respect to corresponding results in \cite{MR4527586,MR3148130,MR2810583}, we only assume that $f$ satisfies the super-linear growth condition at the origin. We believe that the methodology developed here can be adapted to study related problems concerning the existence of solutions for Schr\"{o}dinger-Poisson system.

♻ ☆ Instantaneous continuous loss of regularity for the SQG equation

Diego Córdoba, Luis Martínez-Zoroa, Wojciech S. Ożański

Given $s\in (3/2,2)$ and $\varepsilon >0$, we construct a compactly supported initial data $\theta_0$ such that $\| \theta_0 \|_{H^s}\leq \varepsilon$ and there exist $T>0$, $c>0$ and a local-in-time solution $\theta$ of the SQG equation that is compactly supported in space, continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$, and, for each $t\in [0,T]$, $ \theta (\cdot ,t ) \in {H^{s/(1+ct)}}$ and $ \theta (\cdot ,t ) \not \in {H^\beta }$ for any $\beta > s/(1+ct)$. Moreover, $\theta$ is unique among all solutions with initial condition $\theta_0$ which belong to $C([0,T];H^{1+\alpha })$ for any $\alpha >0$ and is continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$.

comment: 33 pages, 2 figures

♻ ☆ Global existence and pointwise decay for nonlinear waves under the null condition

Shi-Zhuo Looi, Mihai Tohaneanu

This paper proves global existence and sharp pointwise decay for solutions to nonlinear wave equations satisfying the semilinear null condition, on a class of three-dimensional, asymptotically flat, and notably, non-stationary spacetimes. We consider nonlinearities satisfying a generalized null condition which does not necessarily retain its structure when commuted with vector fields. For sufficiently small initial data, and under the assumption that the underlying linear operator satisfies an integrated local energy decay estimate, we prove that solutions exist for all time and we establish sharp pointwise decay estimates for the solution $\phi$ and its vector-fields. The solution itself decays as $|\phi(t,x)| \lesssim \langle t+r \rangle^{-1} \langle t-r \rangle^{-1}$. This rate matches that of the linear equation on a flat background. This rate is sharp, as this behavior holds already for certain time-dependent perturbations of the classical null form on Minkowski space, which we specify.

comment: 38 pages. To appear in Dynamics of PDE

♻ ☆ Finite energy well-posedness for nonlinear Schrödinger equations with non-vanishing conditions at infinity

Paolo Antonelli, Lars Eric Hientzsch, Pierangelo Marcati

Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for energy-subcritical nonlinearities, merely satisfying Kato-type assumptions, is proven, providing the analogue of the well-established local $H^1$-theory for solutions vanishing at infinity. The critical nonlinearity will be simply a byproduct of our analysis and the existing literature. Under an assumption that prevents the onset of a Benjamin-Feir type instability, global well-posedness in the energy space is proven for a) non-negative Hamiltonians, b) sign-indefinite Hamiltonians under additional assumptions on the zeros of the nonlinearity, c) generic nonlinearities and small initial data. The cases b) and c) only concern the 3D case

comment: Author accepted manuscript (AAM)

☆ Analogues of $s$-potential and $s$-energy of mass distribution on Cantor dyadic group and their relation to Hausdorff dimension

Valentin Skvortsov

We introduce an analogue of Riesz $s$-potetial and $s$-energy, $0

comment: 10 pages

☆ Approximating the operator norm of local Hamiltonians via few quantum states

Lars Becker, Joseph Slote, Alexander Volberg, Haonan Zhang

Consider a Hermitian operator $A$ acting on a complex Hilbert space of dimension $2^n$. We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian, its operator norm can be approximated independently of $n$ by maximizing $|\braket{\psi|A|\psi}|$ over a small collection $\mathbf{X}_n$ of product states $\ket{\psi}\in (\mathbf{C}^{2})^{\otimes n}$. More precisely, we show that whenever $A$ is $d$-local, \textit{i.e.,} $\deg(A)\le d$, we have the following discretization-type inequality: \[ \|A\|\le C(d)\max_{\psi\in \mathbf{X}_n}|\braket{\psi|A|\psi}|. \] The constant $C(d)$ depends only on $d$. This collection $\mathbf{X}_n$ of $\psi$'s, termed a \emph{quantum norm design}, is independent of $A$, and consists of product states, and can have cardinality as small as $(1+\eps)^n$, which is essentially tight. Previously, norm designs were known only for homogeneous $d$-localHamiltonians $A$ \cite{L,BGKT,ACKK}, and for non-homogeneous $2$-local traceless $A$ \cite{BGKT}. Several other results, such as boundedness of Rademacher projections for all levels and estimates of operator norms of random Hamiltonians, are also given.

comment: 23 pages

☆ Spectral theory for semigroups on locally convex spaces

Karsten Kruse

In this paper we provide spectral inclusion and mapping theorems for strongly continuous locally equicontinuous semigroups on Hausdorff locally convex spaces. Our results extend the classical spectral inclusion and mapping theorems for strongly continuous semigroups on Banach spaces.

☆ Frame redundancy and Beurling density

Marcin Bownik, Jordy Timo van Velthoven

We show that the frame measure function of a frame in certain reproducing kernel Hilbert spaces on metric measure spaces is given by the reciprocal of the Beurling density of its index set. In addition, we show that each such frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large class of infinite frames. In addition, it shows that the necessary density conditions for sampling in reproducing kernel Hilbert spaces obtained by F\"uhr, Gr\"ochenig, Haimi, Klotz and Romero are optimal. As an application, we also settle the open questions of the existence of frames near the critical density for exponential frames on unbounded sets and for nonlocalized Gabor frames. The techniques used in this paper combine a selector form of Weaver's conjecture and various methods for quantifying the overcompleteness of frames.

☆ Some dynamical properties of weighted shifts on sequence spaces

Michal Hevessy, Tomáš Raunig

Motivated by three recent open questions in the study of linear dynamics, we study weighted shifts on sequence spaces. First, we provide an~example of a~weighted shift on a~locally convex space whose topology is generated by a~sequence of complete seminorms which is generalized hyperbolic, but does not have the shadowing property. Next, we characterise uniform topological expansivity on Fr\'echet spaces satisfying some very natural conditions. Finally, we study the periodic shadowing property on normed spaces leading to a~condition formulated purely in terms of weights which we show is necessary for the periodic shadowing property on $\ell_p$ and equivalent on $c_0$.

comment: 16 pages

☆ Quantization Errors, Human--AI Interaction, and Approximate Fixed Points in $L^1(μ)$

Faruk Alpay, Hamdi Alakkad

We develop a rigorous measure-theoretic framework for the analysis of fixed points of nonexpansive maps in the space $L^1(\mu)$, with explicit consideration of quantization errors arising in fixed-point arithmetic. Our central result shows that every bounded, closed, convex subset of $L^1(\mu)$ that is compact in the topology of local convergence in measure (a property we refer to as measure-compactness) enjoys the fixed point property for nonexpansive mappings. The proof relies on techniques from uniform integrability, convexity in measure, and normal structure theory, including an application of Kirk's theorem. We further analyze the effect of quantization by modeling fixed-point arithmetic as a perturbation of a nonexpansive map, establishing the existence of approximate fixed points under measure-compactness conditions. We also present counterexamples that illustrate the optimality of our assumptions. Beyond the theoretical development, we apply this framework to a human-in-the-loop co-editing system. By formulating the interaction between an AI-generated proposal, a human editor, and a quantizer as a composition of nonexpansive maps on a measure-compact set, we demonstrate the existence of a "stable consensus artefact". We prove that such a consensus state remains an approximate fixed point even under bounded quantization errors, and we provide a concrete example of a human-AI editing loop that fits this framework. Our results underscore the value of measure-theoretic compactness in the design and verification of reliable collaborative systems involving humans and artificial agents.

comment: 18 pages

☆ Une caractérisation de $\mathbb{B}_{\mathrm{dR}}^+$ et $\mathcal{O}\mathbb{B}_{\mathrm{dR}}^+$

Pierre Colmez

We show that ${\mathbf B}_{\mathrm{dR}}^+$ is the universal thickening of ${\mathbf C}_p$. More generally, we show that, if $S$ is a reduced affinoid algebra, $\mathcal{O}\mathbb{B}_{\mathrm{dR}}^+(\overline{S})$ is the universal $S$-thickening of the completion of $\overline{S}$.

comment: 4 pages. In french

☆ Lattice isomorphic Banach lattices of polynomials

Christopher Boyd, Vinícius Miranda

We study D\'iaz-Dineen's problem for regular homogeneous vector-valued polynomials. In particular, we prove that whenever $E^*$ and $F^*$ are lattice isomorphic with at least one having order continuous norm, then $\mathcal{P}^r(^n E; G^*)$ and $\mathcal{P}^r(^n E; G^*)$ are lattice isomorphic for every $n\in \N$ and every Banach lattice $G$. We also study the analogous problem for the classes of regular compact, regular weakly compact, orthogonally additive and regular nuclear polynomials.

comment: 14 pages

☆ On Notions of Expansivity for Operators on Locally Convex Spaces

Nilson C. Bernardes Jr., Félix Martínez-Giménez, Francisco Rodenas

We extend the concept of average expansivity for operators on Banach spaces to operators on arbitrary locally convex spaces. We obtain complete characterizations of the average expansive weighted shifts on Fr\'echet sequence spaces. Moreover, we give a partial answer to a problem proposed by Bernardes et al. in J. Funct. Anal. 288 (2025), Paper No. 110696, by obtaining complete characterizations of the uniformly expansive weighted shifts on K\"othe sequence spaces. Some general properties of the various concepts of expansivity in linear dynamics and several concrete examples are also presented.

♻ ☆ Hyperrigidity II: $R$-dilations, ideals and decompositions

Paweł Pietrzycki, Jan Stochel

We investigate the hyperrigidity of subsets of unital $C^*$-algebras annihilated by states (or, more generally, by completely positive maps). This is closely related to the concept of rigidity at $0$ introduced by G. Salomon, who studied hyperrigid subsets of Cuntz and Cuntz-Krieger algebras. The absence of the unit in a hyperrigid set allows for the existence of $R$-dilations with non-isometric $R$. The existence of such an $R$-dilation forces the state annihilating the hyperrigid set to be a character. Using a dilation-theoretic approach, we provide multiple equivalent criteria for hyperrigidity involving intertwining relations for representations, valid in both commutative and noncommutative settings. We develop structural models for such dilations via orthogonal decompositions into two or three components, determined by defect operators and generalized eigenspaces associated with underlying representations.

♻ ☆ Interior Spectral Windows and Transport for Discrete Fractional Laplacians on $d$-Dimensional Hypercubic Lattices

Nassim Athmouni

We study anisotropic fractional discrete Laplacians $\Delta_{\mathbb{Z}^d}^{\vec{\mathbf{r}}}$ with exponents $\vec{\mathbf{r}}\in\mathbb{R}^d\setminus\{0\}$ on $\ell^2(\mathbb{Z}^d)$. We establish a Mourre estimate on compact energy intervals away from thresholds. As consequences we derive a Limiting Absorption Principle in weighted spaces, propagation estimates (minimal velocity and local decay), and the existence and completeness of local wave operators for perturbations $H=\Delta_{\mathbb{Z}^d}^{\vec{\mathbf{r}}}+W(Q)$, where $W$ is an anisotropically decaying potential of long--range type. In the stationary scattering framework we construct the on--shell scattering matrix $S(\lambda)$, prove the optical theorem, and, under a standard trace--class assumption on $W$, establish the Birman--Krein formula $\det S(\lambda)=\exp(-2\pi i\,\xi(\lambda))$.

♻ ☆ Ergodicity and mixing for locally monotone stochastic evolution equations

Gerardo Barrera, Jonas M. Tölle

We establish general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated exponentially ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the $e$-property of the semigroup. Furthermore, we provide quantitative upper bounds for the Wasserstein $\varepsilon$-mixing times. Examples on possibly unbounded domains include the stochastic incompressible 2D Navier-Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic heat equation, as well as, stochastic semilinear equations such as the 1D stochastic Burgers equation.

comment: 44 pages, 98 references; major revision

♻ ☆ Criterion for the absolute continuity of curves in metric spaces

V. I. Bakhtin

It is proved that a parameterized curve in a metric space $X$ is absolutely continuous if and only if its composition with any Lipschitz function on $X$ is absolutely continuous.

comment: 7 pages; some priority comments and references added

♻ ☆ The Muckenhoupt condition

Zoe Nieraeth

The goal of this paper is to unify the theory of weights beyond the setting of weighted Lebesgue spaces in the general setting of quasi-Banach function spaces. We prove new characterizations for the boundedness of singular integrals, pose several conjectures, and prove partial results related to the duality of the Hardy-Littlewood maximal operator. Furthermore, we give an overview of the theory applied to weighted variable Lebesgue, Morrey, and Musielak-Orlicz spaces.

comment: 46 pages, final version, accepted for publication in J. Funct. Anal

♻ ☆ Contractive projections, conditional expectations, and idempotent coefficient multipliers on $H^p$ spaces $(0

Xiangdi Fu, Kunyu Guo, Dilong Li

In this paper, we investigate contractive projections, conditional expectations, and idempotent coefficient multipliers on the Hardy spaces $H^p(\mathbb{T})$ for $0

comment: We simplify the language

♻ ☆ Extension of contractive projections

Xiangdi Fu, Kunyu Guo, Dilong Li

Through the establishment of several extension theorems, we provide explicit expressions for all contractive projections and 1-complemented subspaces in the Hardy space $H^p(\mathbb{T})$ for $1\leq p<\infty$, $p\neq 2$. Our characterization leads to two corollaries: first, all nontrivial 1-complemented subspaces of $H^p(\mathbb{T})$ are isometric to $H^p(\mathbb{T})$; second, all contractive projections on $H^p(\mathbb{T})$ are restrictions of contractive projections on $L^p(\mathbb{T})$ that leave $H^p(\mathbb{T})$ invariant. The first corollary provides examples of prime Banach spaces \emph{in the isometric sense}, while the second answers a question posed by P. Wojtaszczyk in 2003.

comment: We simplify the language and the proof of Lemma 3.2

♻ ☆ Deformations of Reproducing Kernel Hilbert Spaces on Homogeneous Subvarieties of the Unit Ball

Yasin Watted

We study the relationships between a subvariety of the open unit ball in the complex $d$-dimensional space $\mathbb{C}^{d}$, the reproducing kernel Hilbert space (RKHS) obtained by restricting the Drury-Arveson space to the variety, and its multiplier algebra. We show that when two such RKHSs are almost isometrically isomorphic as RKHSs, their multiplier algebras are likewise almost completely isometrically isomorphic as multiplier algebras. In such cases, the underlying varieties are almost automorphically equivalent. For tractable homogeneous varieties, we further show that if the corresponding multiplier algebras are almost completely isometrically isomorphic as multiplier algebras or one variety is almost the image of the other under a unitary transformation, then the associated RKHSs are almost isometrically isomorphic as RKHSs.

comment: This paper is based on the research thesis entitled "Deformations of Reproducing Kernel Hilbert Spaces on Homogeneous Varieties". 48 pages

☆ Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators

Eddygledson Souza Gama, Francisco Martín, Niels Martin Møller

We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit tangent planes. Nonetheless, we prove that as time $t\to\pm \infty$, any finite entropy, finite genus, embedded, collapsed translating soliton in $\mathbb{R}^3$ converges to a uniquely determined collection of planes. This requires global analysis of quasilinear soliton equations with non-perturbative drifts, which we analyze via sharp non-standard elliptic decay estimates for the drift Laplacian, implying improvements on the Evans-Spruck and Ecker-Huisken estimates in the soliton setting, and exploiting a link from potential theory of the Yukawa equation to heat flows with $L^\infty$-data on non-compact slice curves of these solitons. The structure theorem follows: such solitons decompose at infinity into standard regions asymptotic to planes or grim reaper cylinders. As one application, we classify collapsed translators of entropy two with empty limits as $t\to +\infty$.

comment: 52 pages, 10 figures

☆ Chow and Rashevskii meet Sobolev

Sergey Kryzhevich, Eugene Stepanov, Dario Trevisan

We prove a weak version of the Chow-Rashevskii theorem for vector fields having only Sobolev regularity and generating suitable flows as selections of solutions to the respective ODEs, for a.e.\ initial datum.

comment: 20 pages

☆ On a class of thin obstacle-type problems for the bi-Laplacian operator

Donatella Danielli, Giovanni Gravina

This paper investigates the regularity of solutions and structural properties of the free boundary for a class of fourth-order elliptic problems with Neumann-type boundary conditions. The singular and degenerate elliptic operators studied naturally emerge from the extension procedure for higher-order fractional powers of the Laplacian, while the choice of non-linearity considered encompasses two-phase boundary obstacle problems as a special case. After establishing local regularity properties of solutions, Almgren- and Monneau-type monotonicity formulas are derived and utilized to carry out a blow-up analysis and prove a stratification result for the free boundary.

☆ Hölder Regularity of Dirichlet Problem For The Complex Monge-Ampère Equation

Yuxuan Hu, Bin Zhou

We study the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudo-convex domain in Cn or a Hermitian manifold. Under the condition that the right-hand side lies in Lp function and the boundary data are H\"older continuous, we prove the global H\"older continuity of the solution.

comment: 16 pages, 0 figures

☆ An inductive approach to stochastic estimates for the $\varphi^{4}_2$-equation with correlated coefficient field

Nicolas Clozeau

We develop an inductive approach to obtaining stochastic estimates for the $\varphi^{4}_2$-equation when the coefficient field is correlated with the driving noise. Our method is based on (infinite-dimensional) Gaussian integration by parts with respect to Wick products of Gaussian random variables (more precisely, mollifications of space-time white noise)

comment: Comments welcome !

☆ Liouville type theorem for a class of quasilinear p-Laplace type equations in the half space

Bao Yu, Yang Zhou

We use the method of vector fields to obtain a Liouville-type theorem for a class of quasilinear p-Laplace type equations with conormal boundary condition in the half space. These p-Laplace type equations are the subcritical case of the Euler-Lagrange equation of the Sobolev trace inequality in the half space.

☆ Infinitely many solutions to a conformally invariant elliptic equation with Choquard-type nonlinearity

Mona Almutairi, Mathew Gluck

The existence of an unbounded sequence of solutions to a conformally invariant elliptic equation having nonlocal critical-power nonlinearity is established. The primary obstacle to establishing existence of solutions is the failure of compactness in the Sobolev embedding. To overcome this obstacle, the problem under consideration is lifted to an equivalent problem on the standard sphere so that the symmetries of the sphere can be leveraged. Two classes of symmetries are considered and for each class of symmetries, an unbounded sequence of solutions to the lifted problem with the prescribed symmetries is produced. One class of symmetries always exists and the corresponding solutions are guaranteed to be sign-changing whenever a suitable relationship between the dimension and the nonlocality parameter holds. The other class of symmetries need not always exist but when it exists, the corresponding solutions are guaranteed to be sign-changing.

☆ Global Strong Solutions to the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Initial Data and Vacuum

Qinghao Lei

This paper investigates the three-dimensional axisymmetric compressible Navier-Stokes equations with slip boundary conditions in a cylindrical domain that excludes the axis. For initial density allowed to vanish, the global existence and large time asymptotic behavior of strong and weak solutions are established, provided the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds. It should be noted that this result is obtained without any restrictions on the size of initial data.

☆ Construction of solutions for a critical elliptic system of Hamiltonian type

Congzheng Xuanyuan, Tingfeng Yuan, Yuxia Guo

We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*}\left\{\begin{aligned} &-\Delta u + V(|y'|,y'') u = v^p, \;\; \text{in} \;\; \mathbb{R}^N,\\ &-\Delta v + V(|y'|,y'') v = u^q, \;\; \text{in} \;\; \mathbb{R}^N,\\ &u, v > 0 , (u,v) \in (\dot{W}^{2,\frac{p+1}{p}} (\mathbb{R}^N) \cap L^2(\mathbb{R}^N)) \times (\dot{W}^{2,\frac{q+1}{q}} (\mathbb{R}^N) \cap L^2(\mathbb{R}^N)) ,\end{aligned}\right. \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$ and $V(|y'|, y'')\not\equiv 0$ is a bounded non-negative function in $\mathbb{R}_+\times \mathbb{R}^{N-2}$, $p,q>1$ satisfying $$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}.$$ By using a finite reduction argument and local Pohozaev identities, under the assumption that $N\geq 5$, $(p,q)$ lies in the certain range and $r^2V(r,y'')$ has a stable critical point, we prove that the above problem has infinitely many solutions whose energy can be made arbitrarily large.

☆ Sketches of Nonuniformly Elliptic Schauder Theory

Cristiana De Filippis

Schauder theory is a basic tool in the study of elliptic and parabolic PDEs, asserting that solutions inherit the regularity of the coefficients. It plays a central role in establishing higher regularity for solutions to a broad class of elliptic problems exhibiting ellipticity, including those involving free boundaries. In the linear setting, Schauder theory dates back to the 1920-30s and is now considered classical. Nonlinear extensions were developed in the 1980s. All these classical results are restricted to uniformly elliptic operators and heavily rely on perturbative techniques - freezing the coefficients and comparing the solution to that of a constant-coefficient problem. However, such methods fail in the nonuniformly elliptic setting, where homogeneous a priori estimates break down and standard iteration arguments no longer apply. Here we give a brief survey on recent progresses including the solution to the longstanding problem of proving the validity of Schauder estimates in the nonlinear, nonuniformly elliptic setting.

comment: Survey article for Proceedings of the 9th ECM

☆ On the planar free elastic flow with small oscillation of curvature

Ben Andrews, Glen Wheeler

The free elastic flow that begins at any curve exists for all time. If the initial curve is an $\omega$-fold covered circle (``$\omega$-circle'') the solution expands self-similarly. Very recently, Miura and the second author showed that (topological) $\omega$-circles that are close to multiply-covered round circles are asymptotically stable under the planar free elastic flow, which means that upon rescaling the rescaled flow converges smoothly to the stationary (in the rescaled setting) $\omega$-circle. Closeness in that work was measured via the derivative of the curvature scalar. In the present paper, we improve this by requiring closeness in terms of the curvature scalar itself. The convergence rate we obtain is sharp.

comment: 12 pages

☆ A new proof on quasilinear Schrödinger equations with prescribed mass and combined nonlinearity

Jianhua Chen, Jijiang Sun, Chenggui Yuan, Jian Zhang

In this work, we study the quasilinear Schr\"{o}dinger equation \begin{equation*} \aligned -\Delta u-\Delta(u^2)u=|u|^{p-2}u+|u|^{q-2}u+\lambda u,\,\, x\in\R^N, \endaligned \end{equation*} under the mass constraint \begin{equation*} \int_{\R^N}|u|^2\text{d}x=a, \end{equation*} where $N\geq2$, $20$ is a given mass and $\lambda$ is a Lagrange multiplier. As a continuation of our previous work (Chen et al., 2025, arXiv:2506.07346v1), we establish some results by means of a suitable change of variables as follows: \begin{itemize} \item[{\bf(i) }] {\bf qualitative analysis of the constrained minimization}\\ For $20$; \end{itemize} \begin{itemize} \item[{\bf(ii)}]{\bf existence of two radial distinct normalized solutions}\\ For $2

☆ Long-Time Dynamics of the 3D Vlasov-Maxwell System with Boundaries

Jin Woo Jang, Chanwoo Kim

We construct global-in-time classical solutions to the nonlinear Vlasov-Maxwell system in a three-dimensional half-space beyond the vacuum scattering regime. Our approach combines the construction of stationary solutions to the associated boundary-value problem with a proof of their asymptotic dynamical stability in $L^\infty$ under small perturbations, providing a new framework for understanding long-time wave-particle interactions in the presence of boundaries and interacting magnetic fields. To the best of our knowledge, this work presents the first construction of asymptotically stable non-vacuum steady states under general perturbations in the full three-dimensional nonlinear Vlasov-Maxwell system.

comment: 69 pages

♻ ☆ Reconstruction of 1-D evolution equations and their initial data from one passive measurement

Ali Feizmohammadi

We study formally determined inverse problems with passive measurements for one dimensional evolution equations where the goal is to simultaneously determine both the initial data as well as the variable coefficients in such an equation from the measurement of its solution at a fixed spatial point for a certain amount of time. This can be considered as a one-dimensional model of widely open inverse problems in photo-acoustic and thermo-acoustic tomography. We provide global uniqueness results for wave and heat equations stated on bounded or unbounded spatial intervals. Contrary to all previous related results on the subject, we do not impose any genericity assumptions on the coefficients or initial data. Our proofs are based on creating suitable links to the well understood spectral theory for 1D Schr\"odinger operators. In particular, in the more challenging case of a bounded spatial domain, our proof for the inverse problem partly relies on the following two ingredients, namely (i) a Paley-Wiener type theorem for Schr\"odinger operators due to Remling \cite{Remling2002SchrdingerOA} and a theorem of Levinson \cite{Levinson1940} on distribution of zeros of entire functions of regular growth that together provide a quantifiable link between support of a compactly supported function and the upper density of its vanishing Schr\"odinger spectral modes and (ii) a result of Gesztesy and Simon \cite{Gesztesy1999InverseSA} on partial data inverse spectral problems for reconstructing an unknown potential in a 1D Schr\"odinger operator from the knowledge of only a fraction of its spectrum.

♻ ☆ From Initial Data to Boundary Layers: Neural Networks for Nonlinear Hyperbolic Conservation Laws

Igor Ciril, Khalil Haddaoui, Yohann Tendero

We address the approximation of entropy solutions to initial-boundary value problems for nonlinear strictly hyperbolic conservation laws using neural networks. A general and systematic framework is introduced for the design of efficient and reliable learning algorithms, combining fast convergence during training with accurate predictions. The methodology that relies on solving a certain relaxed related problem is assessed through a series of one-dimensional scalar test cases. These numerical experiments demonstrate the potential of the methodology developed in this paper and its applicability to more complex industrial scenarios.

♻ ☆ On the critical behavior for the semilinear biharmonic heat equation with forcing term in an exterior domain

Nurdaulet N. Tobakhanov, Berikbol T. Torebek

In this paper, we investigate the critical behavior of solutions to the semilinear biharmonic heat equation with forcing term $f(x),$ under six homogeneous boundary conditions. This paper is the first since the seminal work by Bandle, Levine, and Zhang [J. Math. Anal. Appl. 251 (2000) 624-648], to focus on the study of critical exponents in exterior problems for semilinear parabolic equations with a forcing term. By employing a method of test functions and comparison principle, we derive the critical exponents $p_{Crit}$ in the sense of Fujita. Moreover, we show that $p_{Crit}=\infty$ if $N=2,3,4$ and $p_{Crit}=\frac{N}{N-4}$ if $N \geq 5$. The impact of the forcing term on the critical behavior of the problem is also of interest, and thus a second critical exponent in the sense of Lee-Ni, depending on the forcing term is introduced. We also discuss the case $f\equiv 0$, and present the finite-time blow-up results and lifespan estimates of solutions for the subcritical and critical cases. The lifespan estimates of solutions are obtained by employing the method proposed by Ikeda and Sobajama in [Nonlinear Anal. 182 (2019) 57-74].

comment: 43 pages

♻ ☆ Variational eigenvalues of the subelliptic $p$-Laplacian

Mukhtar Karazym

We use the Lusternik-Schnirelman theory to prove the existence of a nondecreasing sequence of variational eigenvalues for the subelliptic $p$-Laplacian subject to the Dirichlet boundary condition.

comment: To appear in Manuscripta Mathematica

♻ ☆ A fast Fourier spectral method for the linearized Boltzmann collision operator

Tianai Yin, Zhenning Cai, Yanli Wang

We introduce a fast Fourier spectral method to compute linearized collision operators of the Boltzmann equation for variable hard-sphere gases. While the state-of-the-art method provides a computational cost O(MN^4 log N), with N being the number of modes in each direction and M being the number of quadrature points on a hemisphere, our method reduces the cost to O(N^4 log N), removing the factor M, which could be large in our numerical tests. The method is applied in a numerical solver for the steady-state Boltzmann equation with quadratic collision operators. Numerical experiments for both spatially homogeneous and inhomogeneous Boltzmann equations have been carried out to test the accuracy and efficiency of our method.

♻ ☆ Stability of axial free-boundary hyperplanes in circular cones

Gian Paolo Leonardi, Giacomo Vianello

Given an axially-symmetric, $(n+1)$-dimensional convex cone $\Omega\subset \mathbb{R}^{n+1}$, we study the stability of the free-boundary minimal surface $\Sigma$ obtained by intersecting $\Omega$ with a $n$-plane that contains the axis of $\Omega$. In the case $n=2$, $\Sigma$ is always unstable, as a special case of the vertex-skipping property that we recently proved in another article. Conversely, as soon as $n \ge 3$ and $\Omega$ has a sufficiently large aperture (depending on the dimension $n$), we show that $\Sigma$ is strictly stable. For our stability analysis, we introduce a Lipschitz flow $\Sigma_{t}[f]$ of deformations of $\Sigma$ associated with a compactly-supported, scalar deformation field $f$, which satisfies the key property $\partial \Sigma_{t}[f] \subset \partial \Omega$ for all $t\in \mathbb{R}$. Then, we compute the lower-right second variation of the area of $\Sigma$ along the flow, and ultimately show that it is positive by exploiting its connection with a functional inequality studied in the context of reaction-diffusion problems.

♻ ☆ Remarks on the commutation relations between the Gauss--Weierstrass semigroup and monomial weights

Yi C. Huang, Ryunosuke Kusaba, Tohru Ozawa

We consider the commutation relations between the Gauss--Weierstrass semigroup on $\mathbb{R}^{n}$, generated by convolution with the complex Gaussian kernel, and monomial weights. We provide an explicit representation and a sharper estimate of the commutation relations with more concise proofs than those in previous works.

comment: 16 pages

♻ ☆ On the positive constant in Arnold's second stability theorem for a bounded domain

Fatao Wang, Guodong Wang, Bijun Zuo

For a steady flow of a two-dimensional ideal fluid, the gradient vectors of the stream function $\psi$ and its vorticity $\omega$ are collinear. Arnold's second stability theorem states that the flow is Lyapunov stable if $0<\nabla\omega/\nabla\psi0$. In this paper, we show that, for a bounded domain, $C_{ar}$ can be taken as the first eigenvalue $\bm\Lambda_1$ of a certain Laplacian eigenvalue problem. When $\nabla\omega/\nabla\psi$ reaches $\bm\Lambda_1$, instability may occur, as illustrated by a non-circular steady flow in a disk; however, a certain form of structural stability still holds. Based on these results, we establish a theorem on the rigidity and orbital stability of steady Euler flows in a disk.

comment: 25 pages; Some writing improvements are provided in this version

☆ Absolutely continuous representing measures of complex sequences

Philipp J. di Dio

In 1989, A. J. Duran [Proc. Amer. Math. Soc. 107 (1989), 731-741] showed, that for every complex sequence $(s_\alpha)_{\alpha\in\mathbb{N}_0^n}$ there exists a Schwartz function $f\in\mathcal{S}(\mathbb{R}^n,\mathbb{C})$ with $\mathrm{supp}\, f\subseteq [0,\infty)^n$ such that $s_\alpha = \int x^\alpha\cdot f(x)~\mathrm{d}x$ for all $\alpha\in\mathbb{N}_0^n$. It has been claimed to be a generalization of the result by T. Sherman [Rend. Circ. Mat. Palermo 13 (1964), 273-278], that every complex sequences is represented by a complex measure on $[0,\infty)^n$. In the present work we use the convolution of sequences and measures to show, that Duran's result is a trivial consequence of Sherman's result. We use our easy proof to extend the Schwartz function result and to show the flexibility in choosing very specific functions $f$.

☆ A note on disjointness and discrete elements in partially ordered vector spaces

Jani Jokela

The notions of disjointness and discrete elements play a prominent role in the classical theory of vector lattices. There are at least three different generalizations of the notion of disjointness to a larger class of partially ordered vector spaces. In recent years, one of these generalizations has been widely studied in the context of pre-Riesz spaces. The notion of $D$-disjointness is the most general of the three disjointness concepts. In this paper we study $D$-disjointness and the related concept of a $D$-discrete element. We establish some basic properties of $D$-discrete elements in Archimedean partially ordered spaces, and we investigate their relationship to discrete elements in the theory of pre-Riesz spaces.

comment: 8 pages

☆ Gohberg-Krupnik Localisation for Discrete Wiener-Hopf Operators on Orlicz Sequence Spaces

Oleksiy Karlovych, Sandra Mary Thampi

Let $\Phi$ be an $N$-function whose Matuszewska-Orlicz indices satisfy $1<\alpha_\Phi\le\beta_\Phi<\infty$. Using these indices, we introduce ``interpolation friendly" classes of Fourier multipliers $M_{[\Phi]}$ and $M_{\langle\Phi\rangle}$ such that $M_{[\Phi]}\subset M_{\langle\Phi\rangle}\subset M_\Phi$, where $M_\Phi$ is the Banach algebra of all Fourier multipliers on the reflexive Orlicz sequence space $\ell^\Phi(\mathbb{Z})$. Applying the Gohberg-Krupnik localisation in the corresponding Calkin algebra, the study of Fredholmness of the discrete Wiener-Hopf operator $T(a)$ with $a\in M_{\langle\Phi\rangle}$ is reduced to that of $T(a_\tau)$ for certain, potentially easier to study, local representatives $a_\tau\in M_{[\Phi]}$ of $a$ at all points $\tau\in[-\pi,\pi)$.

comment: 28 pages

☆ The closure of derivative tent spaces in the logarithmic Bloch-type norm

Rong Yang, Xiangling Zhu

In this paper, the derivative tent space \(DT_p^q(\alpha)\) is introduced. Then, we study \(\mathcal{C}_{\mathcal{B}_{{\log}^\gamma}^\beta}(DT_p^q(\alpha)\cap\mathcal{B}_{{\log}^\gamma}^\beta)\), the closure of the derivative tent space \(DT_p^q(\alpha)\) in the logarithmic Bloch-type space \(\Blog\). As a byproduct, some new characterizations for \(C_\mathcal{B}(\mathcal{D}^p_{\alpha} \cap \mathcal{B})\) and \(C_{\mathcal{B}_{{\log}}}(\mathcal{D}^2_{\alpha}\cap\mathcal{B}_{{\log}})\) are obtained.

☆ Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning

Jia-Qi Yang, Lei Shi

We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which admit discrete spectral decompositions, and (ii) diagonal kernels of the form $K(x,x')=k(x,x')T$, where $k$ is a scalar-valued kernel and $T$ is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbert spaces (RKHSs) that generalize the classical $K=kI$ paradigm, thereby enabling rich structural modeling with rigorous theoretical guarantees. To address target operators lying outside the RKHS, we introduce vector-valued interpolation spaces to precisely quantify misspecification error. Within this framework, we establish dimension-free polynomial convergence rates, demonstrating that nonlinear operator learning can overcome the curse of dimensionality. The use of general operator-valued kernels further allows us to derive rates for intrinsically nonlinear operator learning, going beyond the linear-type behavior inherent in diagonal constructions of $K=kI$. Importantly, this framework accommodates a wide range of operator learning tasks, ranging from integral operators such as Fredholm operators to architectures based on encoder-decoder representations. Moreover, we validate its effectiveness through numerical experiments on the two-dimensional Navier-Stokes equations.

comment: 34 pages, 3 figures

♻ ☆ Convergence of Random Products of Countably Infinitely Many Projections

Rasoul Eskandari, Mohammad Sal Moslehian

Let $r \in \mathbb{N}\cup\{\infty\}$ be a fixed number and let $P_j\,\, (1 \leq j\leq r )$ be the projection onto the closed subspace $\mathcal{M}_j$ of $\mathscr{H}$. We are interested in studying the sequence $P_{i_1}, P_{i_2}, \ldots \in\{P_1, \ldots, P_r\}$. A significant problem is to demonstrate conditions under which the sequence $\{P_{i_n}\cdots P_{i_2}P_{i_1}x\}_{n=1}^\infty$ converges strongly or weakly to $Px$ for any $x\in\mathscr{H}$, where $P$ is the projection onto the intersection $\mathcal{M}=\mathcal{M}_1\cap \ldots \cap \mathcal{M}_r$. Several mathematicians have presented their insights on this matter since von Neumann established his result in the case of $r=2$. In this paper, we give an affirmative answer to a question posed by M. Sakai. We present a result concerning random products of countably infinitely many projections (the case $r=\infty$) incorporating the notion of pseudo-periodic function.

comment: Additional materials and corrected results, 14 pages

♻ ☆ Completeness of the space of absolutely and upper integrable functions with values in a semi-normed space

Rodolfo E. Maza

This paper studies absolute integrability for functions with values in semi- normed spaces and in locally convex topological vector spaces (LCTVS). We introduce an \emph{upper-integral} approach (based on a $\rho$-variational measure $\mu_{\rho}$) to define the spaces $\mathcal{U}^p_{\rho}$ of upper integrable functions and investigate their functional-analytic properties. The main contributions are: \begin{itemize} \item the precise construction of the $\rho$-upper-integrability spaces $\mathcal{U}^p_{\rho}(A;X)$ (and their Fr\'echet analogues), together with the natural semi-norms $\|\cdot\|_{\mathcal{U}^p_{\rho}}$; \item measure-style inequalities adapted to the variational measure $\mu_{\rho}$ (monotone continuity for ascending sets, Fatou-type lemma, and Chebyshev inequality) within the $\rho$-upper-integral framework; \item functional-analytic results: sequential completeness of $\mathcal{U}^p_{\rho}([a,b];X)$ when $X$ is sequentially complete (semi-normed case), and sequential completeness of $\mathcal{U}^p([a,b];X)$ when $X$ is a sequentially complete Fr\'echet space; and \item the closedness of the absolutely integrable subspace $L^p_{\rho}([a,b];X)$ inside $\mathcal{U}^p_{\rho}([a,b];X)$ (hence $L^p([a,b];X)$ is a closed Fr\'echet subspace of $\mathcal{U}^p([a,b];X)$ under the usual hypotheses). \end{itemize}

♻ ☆ Nowhere monotone Riemann integrable derivative without local extrema

Nikolay A. Gusev, Andrey E. Komagorov

We construct a nowhere monotone Riemann integrable derivative which has no local extrema. We also construct a differentiable function $G$ such that $G'$ is Riemann integrable and has an isolated local extrema which is not an inflection point of $G$.

comment: 12 pages, 2 figures (in Russian)

♻ ☆ Wiener's Tauberian theorem in classical and quantum harmonic analysis

Robert Fulsche, Franz Luef, Reinhard F. Werner

We investigate Wiener's Tauberian theorem from the perspective of limit functions, which results in several new versions of the Tauberian theorem. Based on this, we formulate and prove analogous Tauberian theorems for operators in the sense of quantum harmonic analysis. Using these results, we characterize the class of slowly oscillating operators and show that this class is strictly larger than the class of compact operators. Finally, we discuss uniform versions of Wiener's Tauberian theorem and its operator analogue and provide an application of this in operator theory.

comment: Several minor improvement throughout the text

♻ ☆ On the limiting distribution of sums of random multiplicative functions

Ofir Gorodetsky, Mo Dick Wong

We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved by applying the martingale central limit theorem to a suitably truncated sum. This truncation is inspired by work of Najnudel, Paquette, Simm and Vu on subcritical holomorphic multiplicative chaos setting, but analysed with a different conditioning argument generalised from Harper's work on fractional moments to circumvent integrability issues at criticality. A significant part of the proof is devoted to the convergence in probability of the associated partial Euler product to a critical multiplicative chaos measure, independent of the mild shift away from the critical line. Our approach to the universality of critical non-Gaussian multiplicative chaos bypasses the barrier analysis with the help of a modified second moment method, and employs a novel argument based on coupling and homogenisation by change of measure, which could be of independent interest.

comment: 46 pages; source file identical to that of version 2, but re-compiled using TeX Live 2023 as temporary fix to wrong references and broken links associated with unintended behaviour of some LaTeX packages

♻ ☆ Remarks on the commutation relations between the Gauss--Weierstrass semigroup and monomial weights

Yi C. Huang, Ryunosuke Kusaba, Tohru Ozawa

We consider the commutation relations between the Gauss--Weierstrass semigroup on $\mathbb{R}^{n}$, generated by convolution with the complex Gaussian kernel, and monomial weights. We provide an explicit representation and a sharper estimate of the commutation relations with more concise proofs than those in previous works.

comment: 16 pages

☆ Approximation in an optimal design problem governed by the heat equation

Kei Matsushima, Tomoyuki Oka

This paper studies a two-material optimal design problem for the time-averaged duality pairing between a (possibly time-dependent) heat source and the weak solution of an initial-boundary value problem for the heat equation with a two-material diffusion coefficient, under a volume constraint. In general, such optimal designs are not guaranteed to exist, and geometric constraints such as the perimeter are required. As an approximation of the problem with an additional perimeter constraint, a material representation based on a level set function, together with a perturbation of the Dirichlet energy, is employed. It is then shown that optimal level set functions exist for the perturbation problem, and the corresponding minimum value converges to that of the elliptic case, thereby elucidating the long-time behavior. Furthermore, two-material domains satisfying this property are also constructed via the nonlinear diffusion-based level set method. In particular, the asymptotic behavior with respect to the perturbation parameter is clarified, and the validity of the approximation is established.

☆ The concentration-compactness principle for Musielak-Orlicz spaces and applications

Ala Eddine Bahrouni, Anouar Bahrouni

This paper extends the Concentration-Compactness Principle to Musielak-Orlicz spaces, working in both bounded and unbounded domains. We show that our results include important special cases like classical Orlicz spaces, variable exponent spaces, double phase spaces, and a new type of double phase problem where the exponents depend on the solution. Using these general results with variational methods, we prove that certain quasilinear equations with critical nonlinear terms have solutions.

☆ Direct reconstruction of anisotropic self-adjoint inclusions in the Calderón problem

Henrik Garde, David Johansson, Thanasis Zacharopoulos

We extend the monotonicity method for direct exact reconstruction of inclusions in the partial data Calder\'on problem, to the case of anisotropic conductivities in any spatial dimension $d\geq 2$. Specifically, from a local Neumann-to-Dirichlet map, we give reconstruction methods of inclusions based on unknown anisotropic self-adjoint perturbations to a known anisotropic conductivity coefficient. A main assumption is a definiteness condition for the perturbations near the outer inclusion boundaries. This additionally provides new insights into the non-uniqueness issues of the anisotropic Calder\'on problem.

☆ Weak Existence and Uniqueness for Super-Brownian Motion with Irregular Drift

Leonid Mytnik, Johanna Weinberger

We establish weak existence and uniqueness for random field solutions of the one-dimensional SPDE \[ d_tX_t = \frac{1}{2}\Delta X_t +h(X_t)+ \sqrt{X_t}\dot{W}, \quad t\geq 0,\] where $\dot{W}$ is space-time white noise and $h$ is a bounded drift with $h(0)\geq 0$. The proof relies on an extension of the duality relation of the super-Brownian motion, which allows us to treat a broad class of admissible drifts, including functions that are non-Lipschitz or discontinuous at zero.

☆ Alexandrov estimates for polynomial operators by determinant majorization

F. Reese Harvey, Kevin R. Payne

We obtain estimates on the supremum, infimum and oscillation of solutions for a wide class of inhomogeneous fully nonlinear elliptic equations on Euclidean domains where the differential operator is an I-central Garding-Dirichlet operator in the sense of Harvey-Lawson (2024). The argument combines two recent results: an Alexandrov estimate of Payne-Redaelli (2025) for locally semiconvex functions based on the area formula and a determinant majorization estimate of Harvey-Lawson (2024). The determinant majorization estimate has as a special case the arithmetic - geometric mean inequality, so the result includes the classical Alexandrov-Bakelman-Pucci estimate for linear operators. A potential theoretic approach is used involving subequation subharmonics and their dual subharmonics. Semiconvex approximation plays a crucial role.

comment: 37 pages

☆ Navier-Stokes Equations with Fractional Dissipation and Associated Doubly Stochastic Yule Cascades

Radu Dascaliuc, Tuan N. Pham, Enrique Thomann, Edward C. Waymire

Parametric regions are identified in terms of the spatial dimension $d$ and the power $\gamma$ of the Laplacian that separate explosive from non-explosive regimes for the self-similar doubly stochastic Yule cascades (DSY) naturally associated with the deterministic fractional Navier-Stokes equations (FNSE) on $\mathbb{R}^d$ in the scaling-supercritical setting. Explosion and/or geometric properties of the DSY, are then used to establish non-uniqueness, local existence, and finite-time blow-up results for a scalar partial differential equation associated to the FNSE through a majorization principle at the level of stochastic solution processes. The solution processes themselves are constructed from the DSY and yield solutions to the FNSE upon taking expectations. In the special case $d=2$, a closed-form expression for the solution process of the FNSE is derived. This representation is employed to prove finite-time loss of integrability of the solution process for sufficiently large initial data. Notably, this lack of integrability does not necessarily imply blow-up of the FNSE solutions themselves. In fact, in the radially symmetric case, solutions can be continued beyond the integrability threshold by employing a modified notion of averaging.

☆ Mixed regularity and sparse grid approximations of $N$-body Schrödinger evolution equation

Long Meng, Dexuan Zhou

In this paper, we present a mathematical analysis of time-dependent $N$-body electronic systems and establish mixed regularity for the corresponding wavefunctions. Based on this, we develop sparse grid approximations to reduce computational complexity, including a sparse grid Gaussian-type orbital (GTO) scheme. We validate the approach on the Helium atom (${\rm He}$) and Hydrogen molecule (${\rm H}_2$), showing that sparse grid GTOs offer an efficient alternative to full grid discretizations.

comment: 20 pages, 4 figures

♻ ☆ Superposition Property in Disjoint Variables for the Infinity Laplace Equation

Qing Liu, Juan J. Manfredi, Xiaodan Zhou

We establish a superposition principle in disjoint variables for the inhomogeneous infinity-Laplace equation. We show that the sum of viscosity solutions of the inhomogeneous infinity-Laplace equation in separate domains is a viscosity solution in the product domain. This result has been used in the literature with certain particular choices of solutions to simplify regularity analysis for a general inhomogeneous infinity-Laplace equation by reducing it to the case without sign-changing inhomogeneous terms and vanishing gradient singularities. We present a proof of this superposition principle for general viscosity solutions. We also explore generalization in metric spaces using cone comparison techniques and study related properties for general elliptic and convex equations.

comment: 12 pages

♻ ☆ Hyperbolic Keller-Segel equations with sensitivity adjustment in Besov spaces: Global well-posedness and blow-up criteria

Bo Bi, Lei Zhang

This paper focuses on the initial value problem for the hyperbolic Keller-Segel (HKS) equation with sensitivity adjustment in Besov sapces over $\mathbb{R}^d$: $\partial_t u + \nabla \cdot \left(\varrho (t) u (1 - u) \nabla S \right)= 0$, $\Delta S = S - u$, where $\varrho (t) (1 - u)$ denotes the adjustment of the classical chemotaxis sensitively by virtue of a time-dependent function $\varrho$. In the case of $\varrho\equiv 1$, existed results [Zhou et al., \emph{J. Differ. Equ.}, 302 (2021) 662-679] and [Zhang et al., \emph{J. Differ. Equ.}, 334 (2022) 451-489] have shown that the HKS equation admits local-in-time Besov solution, whereas the global theory in Besov space still remains an unsolved problem. The main novelty of our observation lies in the fact that if the chemotaxis sensitivity is adjusted by the function $\varrho(t)$ with suitable integrability over $[0,\infty)$, then the associated HKS equation possesses a unique global-in-time Besov solution. As an application, we conclude that the HKS equation with weakly dissipation $-\lambda u$ (i.e., a nonzero interaction-related source term) can also be globally solved in the framework of Besov spaces. Moreover, we derive two types of blow-up criteria for strong solutions in both critical and non-critical Besov spaces, and explicitly characterize the lower bound of blow-up time. These findings reveals how time-dependent parameters (especially, the dissipation parameter $\lambda$) affect the global existence of solutions.

♻ ☆ A new monotonicity formula for quasilinear elliptic free boundary problems

Aram Karakhanyan

We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional \[ J(u)=\int_\Omega F(|\nabla u|^2)+\mbox{meas}(\{u>0\}\cap \Omega), \] where $F$ is a density function satisfying some structural conditions. The onus of proof lies with the careful analysis of the ghost function, the gradient part in the Helmholtz-W\'eyl decomposition of a nonlinear flux that appears in the domain variation formula for $J(u)$. As an application we prove full regularity for a class of quasilinear Bernoulli type free boundary problems in $\R^3$.

☆ Actions of Fell bundles

Erik Bédos, Roberto COnti

We introduce and study actions of Fell bundles over discrete groups on Hilbert bundles. Many examples of such actions are presented. We discuss the connection with positive definite bundle maps between Fell bundles, culminating in the unital case in a Gelfand-Raikov type theorem. We also use these actions to construct C*-correspondences over cross-sectional C*-algebras of Fell bundles.

comment: 47 pages

☆ Algebra bundles, projective flatness and rationally-deformed tori

Alexandru Chirvasitu

We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the \v{C}ech cohomology group $H^2(\mathbb{T},\mu_q:=\text{$q^{th}$ roots of unity})$ (respectively $H^2(\mathbb{Z})$) via the image of $[A]\in H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))$ through $H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))\xrightarrow{\quad}H^2(\mathbb{T},\mu(q,\mathcal{C}_{\mathbb{T}}))$ (respectively the first Chern class $c_1(\mathcal{E})$). This is in the spirit of Auslander-Szczarba's result identifying real flat bundles on the torus with their first two Stiefel-Whitney classes, and contrasts with classifying spaces $B\Gamma$ of compact Lie groups $\Gamma$ (as opposed to $\mathbb{T}^n\cong B\mathbb{Z}^n$), on which flat non-trivial vector bundles abound. The discussion both recovers the Disney-Elliott-Kumjian-Raeburn classification of rational non-commutative tori $\mathbb{T}^n_{\theta}$ with a different, bundle-theoretic proof, and sheds some light on the connection between topological invariants associated to $\mathbb{T}^2_{\theta}$, $\theta\in\mathbb{Q}$ by Rieffel and respectively H{\o}egh-Krohn-Skjelbred.

comment: 16 pages + references

☆ A study of ferronematic thin films including a stray field energy

Shilpa Dutta, James Dalby, Apala Majumdar, Anja Schlömerkemper

Ferronematic materials are colloidal suspensions of magnetic particles in liquid crystals. They are complex materials with potential applications in display technologies, sensors, microfluidics devices, etc. We consider a model for ferronematics in a 2D domain with a variational approach. The proposed free energy of the ferronematic system depends on the Landau-de Gennes (LdG) order parameter $\mathbf{Q}$ and the magnetization $\mathbf{M}$, and incorporates the complex interaction between the liquid crystal molecules and the magnetic particles in the presence of an external magnetic field $\mathbf{H}_{ext}$. The energy functional combines the Landau-de Gennes nematic energy density and energy densities from the theory of micromagnetics including (an approximation of) the stray field energy and energetic contributions from an external magnetic field. For the proposed ferronematic energy, we first prove the existence of an energy minimizer and then the uniqueness of the minimizer in certain parameter regimes. Secondly, we numerically compute stable ferronematic equilibria by solving the gradient flow equations associated with the proposed ferronematic energy. The numerical results show that the stray field influences the localization of the interior nematic defects and magnetic vortices.

comment: 28 pages, 9 figures

☆ Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE

Dominic Blanco, Matthieu Cadiot

In this article, we extend the framework developed previously to allow for rigorous proofs of existence of smooth, localized solutions in semi-linear partial differential equations possessing both space and non-space group symmetries. We demonstrate our approach on the Swift-Hohenberg model. In particular, for a given symmetry group $\mathcal{G}$, we construct a natural Hilbert space $H^l_{\mathcal{G}}$ containing only functions with $\mathcal{G}$-symmetry. In this space, products and differential operators are well-defined allowing for the study of autonomous semi-linear PDEs. Depending on the properties of $\mathcal{G}$, we derive a Newton-Kantorovich approach based on the construction of an approximate inverse around an approximate solution, $u_0$. More specifically, combining a meticulous analysis and computer-assisted techniques, the Newton-Kantorovich approach is validated thanks to the computation of some explicit bounds. The strategy for constructing $u_0$, the approximate inverse, and the computation of these bounds will depend on the properties of $\mathcal{G}$. We demonstrate the methodology on the 2D Swift-Hohenberg PDE by proving the existence of various dihedral localized patterns. The algorithmic details to perform the computer-assisted proofs can be found on Github.

☆ Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on $\partial_t A$

Martin Ulmer

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume an $L^1$-Carleson condition on only $|\partial_t A|$ the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this $L^1$-Carleson condition on $|\partial_t A|$.

☆ Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves

Roberto Feola, Riccardo Montalto, Shulamit Terracina

Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics (which is a widely open problem). In this paper we prove the existence of time quasi-periodic traveling wave solutions for three-dimensional pure gravity water waves in finite depth, on flat tori, with an arbitrary number of speeds of propagation. These solutions are global in time, they do not reduce to stationary solutions in any moving reference frame and they are approximately given by finite sums of Stokes waves traveling with rationally independent speeds of propagation. This is a very hard small divisors problem for Partial Differential Equations due to the fact that one deals with a dispersive quasi-linear PDE in higher dimension with a very complicated geometry of the resonances. Our result is the first KAM (Kolmogorov-Arnold-Moser) result for an autonomous, dispersive, quasi-linear PDE in dimension greater than one and it is the first example of global solutions, which do not reduce to steady ones in any moving reference frame, for 3D water waves equations on compact domains.

comment: 206 pages. Keywords: Fluid Mechanics, Water Waves, quasi-periodic traveling waves, Microlocal Analysis, Small Divisors

☆ Long-time behavior of a nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potentials

Maoyin Lv, Hao Wu

We investigate the long-time behavior of a nonlocal Cahn--Hilliard equation in a bounded domain $\Omega\subset\mathbb{R}^d$ $(d=2,3)$, subject to a kinetic rate dependent nonlocal dynamic boundary condition. The kinetic rate $1/L$, with $L\in[0,+\infty)$, distinguishes different types of bulk-surface interactions. When $L\in[0,+\infty)$, for a general class of singular potentials including the physically relevant logarithmic potential, we establish the existence of a global attractor $\mathcal{A}_m^L$ in a suitable complete metric space. Moreover, we verify that the global attractor $\mathcal{A}_m^0$ is stable with respect to perturbations $\mathcal{A}_m^L$ for small $L>0$. For the case $L\in(0,+\infty)$, based on the strict separation property of solutions, we prove the existence of exponential attractors through a short trajectory type technique, which also yields that the global attractor has finite fractal dimension. Finally, when $L\in(0,+\infty)$, by usage of a generalized {\L}ojasiewicz-Simon inequality and an Alikakos-Moser type iteration, we show that every global weak solution converges to a single equilibrium in $\mathcal{L}^\infty$ as time tends to infinity.

☆ Anomalous dissipation and regularization in isotropic Gaussian turbulence

Theodore D. Drivas, Lucio Galeati, Umberto Pappalettera

In this work we rigorously establish a number of properties of "turbulent" solutions to the stochastic transport and the stochastic continuity equations constructed by Le Jan and Raimond in [Ann. Probab. 30(2): 826-873, 2002]. The advecting velocity field, not necessarily incompressible, is Gaussian and white-in-time, space-homogeneous and isotropic, with $\alpha$-H\"older regularity in space, $\alpha\in (0,1)$. We cover the full range of compressibility ratios giving spontaneous stochasticity of particle trajectories. For the stochastic transport equation, we prove that generic $L^2_x$ data experience anomalous dissipation of the mean energy, and study basic properties of the resulting anomalous dissipation measure. Moreover, we show that starting from such irregular data, the solution immediately gains regularity and enters into a fractional Sobolev space $H^{1-\alpha-}_x$. The proof of the latter is obtained as a consequence of a new sharp regularity result for the degenerate parabolic PDE satisfied by the associated two-point self-correlation function, which is of independent interest. In the incompressible case, a Duchon-Robert-type formula for the anomalous dissipation measure is derived, making a precise connection between this self-regularizing effect and a limit on the flux of energy in the turbulent cascade. Finally, for the stochastic continuity equation, we prove that solutions starting from a Dirac delta initial condition undergo an average squared dispersion growing with respect to time as $t^{1/(1-\alpha)}$, rigorously establishing the analogue of Richardson's law of particle separations in fluid dynamics.

comment: 68 pages. All comments are welcome!

☆ Subordinators and time-space fractional diffusion equations

Mohamed Majdoub, Ezzedine Mliki

We study the long-time behavior of solutions to a class of evolution equations arising from random-time changes driven by subordinators. Our focus is on fractional diffusion equations involving mixed local and nonlocal operators. By combining techniques from probability theory, asymptotic analysis, and partial differential equations (PDEs), we characterize the dynamics of the subordinated solutions. This approach extends classical fractional dynamics and establishes a deeper connection between stochastic processes and deterministic PDEs.

comment: 15 pages

☆ The isoperimetric inequality for the capillary energy outside convex sets

N. Fusco, V. Julin, M. Morini, A. Pratelli

We study the isoperimetric problem for capillary hypersurfaces with a general contact angle $\theta \in (0, \pi)$, outside arbitrary convex sets. We prove that the capillary energy of any surface supported on any such convex set is larger than that of a spherical cap with the same volume and the same contact angle on a flat support, and we characterize the equality cases. This provides a complete solution to the isoperimetric problem for capillary surfaces outside convex sets at arbitrary contact angles, generalizing the well-known Choe-Ghomi-Ritor\'e inequality, which corresponds to the case $\theta=\frac\pi2$.

comment: This article supersedes the earlier version arXiv:2406.19011

☆ A Spectral Localization Method for Time-Fractional Integro-Differential Equations with Nonsmooth Data

Lijing Zhao, Rui Zhao, Wenyi Tian, Yufeng Nie

In this work, we develop a localized numerical scheme with low regularity requirements for solving time-fractional integro-differential equations. First, a fully discrete numerical scheme is constructed. Specifically, for temporal discretization, we employ the contour integral method (CIM) with parameterized hyperbolic contours to approximate the nonlocal operators. For spatial discretization, the standard piecewise linear Galerkin finite element method (FEM) is used. We then provide a rigorous error analysis, demonstrating that the proposed scheme achieves high accuracy even for problems with nonsmooth/vanishing initial values or low-regularity solutions, featuring spectral accuracy in time and second-order convergence in space. Finally, a series of numerical experiments in both 1-D and 2-D validate the theoretical findings and confirm that the algorithm combines the advantages of spectral accuracy, low computational cost, and efficient memory usage.

☆ From the Klein-Gordon Equation to the Relativistic Quantum Hydrodynamic System: Local Well-posedness

Ben Duan, Jun Li, Bin Guo, Rongrong Yan

In the Klein-Gordon equation, the quantum and relativistic parameters are intricately coupled, which complicates the direct consideration of quantum fluctuations. In this paper, the so-called Relativistic Quantum Hydrodynamics System is derived from the Klein-Gordon equation with Poisson effects via the Madelung transformation, providing a fresh perspective for analyzing the singular limits, such as the semi-classical limits and non-relativistic limits. The Relativistic Quantum Hydrodynamics System, when the semiclassical limit is taken, formally reduces to the Relativistic Hydrodynamics System. When the relativistic limit is taken, it formally reduces to the Quantum Hydrodynamics System. Additionally, we establish the local classical solutions for the Cauchy problem associated with the Relativistic Quantum Hydrodynamic System. The initial density value is assumed to be a small perturbation of some constant state, but the other initial values do not require this restriction. The key point is that the Relativistic Quantum Hydrodynamic System is reformulated as a hyperbolic-elliptic coupled system.

☆ Blow-up criteria for the semilinear parabolic equations driven by mixed local-nonlocal operators

Vishvesh Kumar, Berikbol T. Torebek

The main goal of this paper is to establish \emph{necessary and sufficient conditions} for the nonexistence of a global solution to the semilinear heat equation with a mixed local--nonlocal operator $ -\Delta + (-\Delta)^\sigma$, under a general time-dependent nonlinearity. Our results complement the recent work of Carhuas-Torre et al. [ArXiv, (2025), arXiv:2505.20401], in which the authors provide sufficient conditions for the existence and nonexistence of global solutions. In particular, our results recover the critical Fujita exponent for time-independent power-type nonlinearities, as obtained by Biagi et al. [Bull. London Math. Soc. (2024), 1--20] and Del Pezzo et al. [Nonlinear Anal. 255 (2025), 113761].

comment: 14 pages

☆ Even Cone Spherical Metrics: Blow-Up at a Cone Singularity

Ting-Jung Kuo, Xuanpu Liang, Ping-Hsiang Wu

We study families of spherical metrics on the flat torus $E_{\tau}$ $=$ $\mathbb{C}/\Lambda_{\tau}$ with blow-up behavior at prescribed conical singularities at $0$ and $\pm p$, where the cone angle at $0$ is $6\pi$, and at $\pm p$ is $4\pi$. We prove that the existence of such a necessarily unique, even family of spherical metrics is completely determined by the geometry of the torus: such a family exists if and only if\textbf{ }the Green function $G(z;\tau)$ admits a pair of nontrivial critical points $\pm a$. In this case, the cone point $p$ must equal $a$, and the corresponding monodromy data is $\left( 2r,2s\right) $, where $a=r+s\tau.$ An explicit transformation relating this family to the one with a single conical singularity of angle $6\pi$ at the origin is established in Theorem 1.4. A rigidity result for rhombic tori is proved in Theorem 1.5.

comment: 26 pages, 1 figure

☆ Normalized solutions to a Choquard equation involving mixed local and nonlocal operators

J. Giacomoni, Nidhi Nidhi, K. Sreenadh

In the present paper, we study the existence of normalized solutions for a Choquard type equation involving mixed diffusion type operators. We also provide regularity results of these solutions. Next, the equivalence between existence of normalized solutions and the existence of normalized ground states is established.

☆ Conditional existence of maximizers for the Tomas-Stein inequality for the sphere

Shuanglin Shao, Ming Wang

The Tomas-Stein inequality for a compact subset $\Gamma$ of the sphere $S^d$ states that the mapping $f\mapsto \widehat{f\sigma}$ is bounded from $L^2(\Gamma,\sigma)$ to $L^{2+4/d}(\R^{d+1})$. Then conditional on a strict comparison between the best constants for the sphere and for the Strichartz inequality for the Schr\"odinger equations, we prove that there exist functions which extremize this inequality, and any extremising sequence has a subsequence which converges to an extremizer. The method is based on the refined Tomas-Stein inequality for the sphere and the profile decompositions. The key ingredient to establish orthogonality in profile decompositions is that we use Tao's sharp bilinear restriction theorem for the paraboloids beyond the Tomas-Stein range. Similar results have been previously established by Frank, Lieb and Sabin \cite{Frank-Lieb-Sabin:2007:maxi-sphere-2d}, where they used the method of the missing mass.

comment: 35 pages. 3 figures

☆ Simultaneous determination of wave speed, diffusivity and nonlinearity in the Westervelt equation using complex time-periodic solutions

Sebastian Acosta, Benjamin Palacios

We consider an inverse problem governed by the Westervelt equation with linear diffusivity and quadratic-type nonlinearity. The objective of this problem is to recover all the coefficients of this nonlinear partial differential equation. We show that, by constructing complex-valued time-periodic solutions excited from the boundary time-harmonically at a sufficiently high frequency, knowledge of the first- and second-harmonic Cauchy data at the boundary is sufficient to simultaneously determine the wave speed, diffusivity and nonlinearity in the interior of the domain of interest.

♻ ☆ Removable sets for pseudoconvexity for weakly smooth boundaries

Quang Dieu Nguyen, Pascal J. Thomas

We show that for bounded domains in $\mathbb C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial \Omega \setminus F$ is $\mathcal C^{2}$-smooth and locally pseudoconvex at every point, then $\Omega$ is globally pseudoconvex. Unlike in the globally $\mathcal C^{2}$-smooth case, the condition ``$F$ of (relative) empty interior'' is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity.

comment: Simplified proof of the result by Grauert & Remmert; various small corrections; 16 pages

♻ ☆ Extensions of Schoen--Simon--Yau and Schoen--Simon theorems via iteration à la De Giorgi

Costante Bellettini

We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of $6$-dimensional (stable minimal) immersions. The key step is an $\epsilon$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension $n\geq 2$, that may have a singular set $\Sigma$ of locally finite $\mathcal{H}^{n-2}$-measure, and that is weakly close to a hyperplane. (In fact, if $\mathcal{H}^{n-2}(\Sigma)=0$, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an $\epsilon$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen--Simon theorem (1981). In both $\epsilon$-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method \`{a} la De Giorgi (from the linear De Giorgi--Nash--Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both $\epsilon$-regularity theorems the method gives explicit constants that quantify the required smallness.)

comment: 34 pp

♻ ☆ Bounds for spectral projectors on the three-dimensional torus

Pierre Germain, Simon L. Rydin Myerson, Daniel Pezzi

We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window limit we are left with $L^p$ norms of eigenfunctions of the Laplacian, as considered for instance by Bourgain. For the three-dimensional torus we prove new cases of a previous conjecture of the first two authors concerning the size of these norms; we also refine certain prior results to remove $\epsilon$-losses in all dimensions. We use methods from number theory: the geometry of numbers, the circle method and exponential sum bounds due to Guo. We complement these techniques with height splitting and a bilinear argument to prove sharp results. We exposit on the various techniques used and their limitations.

comment: 51 pages, 1 figure. Slightly updated introduction and theorem numbering. Results unchanged

♻ ☆ Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity

Serhii D. Koval, Elsa Dos Santos Cardoso-Bihlo, Roman O. Popovych

Using the original advanced version of the direct method, we efficiently compute the equivalence groupoids and equivalence groups of two peculiar classes of Kolmogorov backward equations with power diffusivity and solve the problems of their complete group classifications. The results on the equivalence groups are double-checked with the algebraic method. Within these classes, the remarkable Fokker-Planck and the fine Kolmogorov backward equations are distinguished by their exceptional symmetry properties. We extend the known results on these two equations to their counterparts with respect to a nontrivial discrete equivalence transformation. Additionally, we carry out Lie reductions of the equations under consideration up to the point equivalence, exhaustively study their hidden Lie symmetries and generate wider families of their new exact solutions via acting by their recursion operators on constructed Lie-invariant solutions. This analysis reveals eight powers of the space variable with exponents -1, 0, 1, 2, 3, 4, 5 and 6 as values of the diffusion coefficient that are prominent due to symmetry properties of the corresponding equations.

comment: 39 pages, published version, minor corrections

♻ ☆ Twisting in One Dimensional Periodic Vlasov-Poisson System

Sagnwook Tae

We prove that twisting and filamentation occur near a family of stable steady states for one dimensional periodic Vlasov-Poisson system, describing the electron dynamics under a fixed ion background. More precisely, we establish the growth in time of the L1 norm of the gradient for the electron distribution function and the corresponding flow map in the phase space. To support this result, we prove existence results of stable steady states for a class of ion densities on the torus.

♻ ☆ Two-phase flows through porous media described by a Cahn--Hilliard--Brinkman model with dynamic boundary conditions

Pierluigi Colli, Patrik Knopf, Giulio Schimperna, Andrea Signori

We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field as well as a convective Cahn--Hilliard equation with dynamic boundary conditions for the phase-field, which describes the location of the two fluids within the domain. The dynamic boundary conditions are incorporated to model the interaction of the fluids with the wall of the container more precisely. In particular, they allow for a dynamic evolution of the contact angle between the interface separating the fluids and the boundary, and also for a convection-induced motion of the corresponding contact line. For our model, we first prove the existence of global-in-time weak solutions in the case where regular potentials are used in the Cahn--Hilliard subsystem. In this case, we can further show the uniqueness of the weak solution under suitable additional assumptions. Moreover, we further prove the existence of weak solutions in the case of singular potentials. Therefore, we regularize such singular potentials by a Yosida approximation, such that the results for regular potentials can be applied, and eventually pass to the limit in this approximation scheme.

♻ ☆ Thermo-Elasticity Problems with Evolving Microstructures

Michael Eden, Adrian Muntean

We consider the mathematical analysis and homogenization of a moving boundary problem posed for a highly heterogeneous, periodically perforated domain. More specifically, we are looking at a one-phase thermo-elasticity system with phase transformations where small inclusions, initially periodically distributed, are growing or shrinking based on a kinetic under-cooling-type law and where surface stresses are created based on the curvature of the phase interface. This growth is assumed to be uniform in each individual cell of the the perforated domain. After transforming to the initial reference configuration (utilizing the Hanzawa transformation), we use the contraction mapping principle to show the existence of a unique solution for a possibly small but $\varespilon$-independent time interval ($\varespilon$ is here the scale of heterogeneity). In the homogenization limit, we discover a macroscopic thermo-elasticity problem which is strongly non-linearly coupled (via an internal parameter called height function) to local changes in geometry. As a direct byproduct of the mathematical analysis work, we present an alternative equivalent formulation which lends itself to an effective precomputing strategy that is very much needed as the limit problem is computationally expensive.

comment: 31 pages

♻ ☆ Functional tilings and the Coven-Meyerowitz tiling conditions

Gergely Kiss, Itay Londner, Máté Matolcsi, Gábor Somlai

Coven and Meyerowitz formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture is reduced to sets which tile a finite cyclic group $\mathbb{Z}_M$. In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions $f,g$, such that $f(0)=g(0)=1$, $f\ast g=\mathbf{1}_{\mathbb{Z}_M}$ is a functional tiling, and $f, g$ satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future.

comment: 20 pages. Several misprints and inaccuracies from the original submission were corrected. More detailed explanation in some proofs are provided

♻ ☆ Singular perturbations models in phase transitions for anisotropic higher-order materials

Giuseppe Cosma Brusca, Davide Donati, Chiara Trifone

We discuss a model for phase transitions in which a double-well potential is singularly perturbed by possibly several terms involving different, arbitrarily high orders of derivation. We study by $\Gamma$-convergence the asymptotic behaviour as $\varepsilon\to 0$ of the functionals \begin{equation*} F_\varepsilon(u):=\int_\Omega \Bigl[\frac{1}{\varepsilon}W(u)+\sum_{\ell=1}^{k}q_\ell\varepsilon^{2\ell-1}|\nabla^{(\ell)}u|_\ell^2\Bigr]\,dx, \qquad u\in H^k(\Omega), \end{equation*} for fixed $k>1$ integer, addressing also to the case in which the coefficients $q_1,...,q_{k-1}$ are negative and $|\cdot|_\ell$ is any norm on the space of symmetric $\ell$-tensors for each $\ell\in\{1,...,k\}$. The negativity of the coefficients leads to the lack of a priori bounds on the functionals; such issue is overcome by proving a nonlinear interpolation inequality. With this inequality at our disposal, a compactness result is achieved by resorting to the recent paper [10]. A further difficulty is the presence of general tensor norms which carry anisotropies, making standard slicing arguments not suitable. We prove that the $\Gamma$-limit is finite only on sharp interfaces and that it equals an anisotropic perimeter, with a surface energy density described by a cell formula.

comment: 41 pages

♻ ☆ Physical Space Proof of Bilinear Estimates and Applications to Nonlinear Dispersive Equations (II)

Xinfeng Hu, Li Tu, Yi Zhou

The work by Kenig-Ponce-Vega [15] initiated the use of Bourgain spaces to study the low-regularity well-posedness of semilinear dispersive equations. Since then, the Bourgain space method has become the dominant, and almost the only method to deal with this problem. The goal of this series of papers is to propose an alternative approach for this problem that does not rely on Bourgain spaces. Our method is based on a bilinear estimate, which is proved in a physical space approach by a new div-curl type lemma introduced by the third author. Combining these ingredients with a Strichartz estimate of mixed spatial integrability, we will illustrate our method in the present paper by reproducing best known local well-posedness results for the 2d and 3d Zakharov system from Bejenaru-Herr-Holmer-Tataru [2] and Bejenaru-Herr [1].

comment: 27 pages

♻ ☆ Quadrature Domains and the Faber Transform

Andrew J. Graven, Nikolai G. Makarov

We present a framework for reconstructing any simply connected, bounded or unbounded, quadrature domain $\Omega$ from its quadrature function $h$. Using the Faber transform, we derive formulae directly relating $h$ to the Riemann map for $\Omega$. Through this approach, we obtain a complete classification of one point quadrature domains with complex charge. We proceed to develop a theory of weighted quadrature domains with respect to weights of the form $\rho_a(w)=|w|^{2(a-1)}$ when $a > 0$ ("power-weighted" quadrature domains) and the limiting case of when $a=0$ ("log-weighted" quadrature domains). Furthermore, we obtain Faber transform formulae for reconstructing weighted quadrature domains from their respective quadrature functions. Several examples are presented throughout to illustrate this approach both in the simply connected setting and in the presence of rotational symmetry.

comment: 60 Pages, 21 Figures

♻ ☆ Quantitative homogenization and large-scale regularity of Poisson point clouds

Scott Armstrong, Raghavendra Venkatraman

We prove quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters--that is, Poisson point clouds with edges connecting points which are closer than some fixed distance. We show that, on large scales, harmonic functions resemble harmonic functions in Euclidean space with sharp quantitative bounds on their difference. In particular, for every point cloud which is supercritical (meaning that the intensity of the Poisson process is larger than the critical parameter which guarantees the existence of an infinite connected component), we obtain optimal corrector bounds, homogenization error estimates and large-scale regularity results.

comment: 62 pages

♻ ☆ Analysis of a parabolic-hyperbolic hybrid population model

Qihua Huang, Minglong Wang, Yixiang Wu

This paper is concerned with the global dynamics of a hybrid parabolic-hyperbolic model describing populations with distinct dispersal and sedentary stages. We first establish the global well-posedness of solutions, prove a comparison principle, and demonstrate the asymptotic smoothness of the solution semiflow. Through the spectral analysis of the linearized system, we derive and characterize the net reproductive rate $\mathcal{R}_{0}$. Furthermore, an explicit relationship between $\mathcal{R}_{0}$ and the principal eigenvalue of the linearized system is analyzed. Under appropriate monotonicity assumptions, we show that $\mathcal{R}_{0}$ serves as a threshold parameter that completely determines the stability of steady states of the system. More precisely, when $\mathcal{R}_{0}<1$, the trivial equilibrium is globally asymptotical stable, while when $\mathcal{R}_{0}>1$, the system is uniformly persistent and there is a positive equilibrium which is unique and globally asymptotical stable.

♻ ☆ Degenerate elliptic equations with $Φ$-admissible weights

Lyudmila Korobenko

We develop regularity theory for degenerate elliptic equations with the degeneracy controlled by a weight. More precisely, we show local boundedness and continuity of weak solutions under the assumption of a weighted Orlicz-Sobolev and Poincar\'{e} inequalities. The proof relies on a modified DeGiorgi iteration scheme, developed in arXiv:1608.01630 and arXiv:1703.00774. The Orlicz-Sobolev inequality we assume here is much weaker than the classical $(2\sigma,2)$ Sobolev inequality with $\sigma>1$ which is typically used in the DeGiorgi or Moser iteration.

comment: some typos fixed

♻ ☆ Twisting in One Dimensional Periodic Vlasov-Poisson System

Sangwook Tae

We prove that twisting and filamentation occur near a family of stable steady states for one dimensional periodic Vlasov-Poisson system, describing the electron dynamics under a fixed ion background. More precisely, we establish the growth in time of the L1 norm of the gradient for the electron distribution function and the corresponding flow map in the phase space. To support this result, we prove existence results of stable steady states for a class of ion densities on the torus.

☆ Entropy and Learning of Lipschitz Functions under Log-Concave Measures

Pierre Bizeul, Boaz Klartag

We study regression of $1$-Lipschitz functions under a log-concave measure $\mu$ on $\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are ineffective. We analyze two polynomial-based procedures: the projection estimator, which relies on knowledge of an orthogonal polynomial basis of $\mu$, and the least-squares estimator over low-degree polynomials, which requires no knowledge of $\mu$ whatsoever. Their risk is governed by the rate of polynomial approximation of Lipschitz functions in $L^2(\mu)$. When this rate matches the Gaussian one, we show that both estimators achieve minimax bounds over a wide range of parameters. A key ingredient is sharp entropy estimates for the class of $1$-Lipschitz functions in $L^2(\mu)$, which are new even in the Gaussian setting.

comment: 45 pages

☆ New Approaches to the Fixed Point Property in L^1 Spaces

Faruk Alpay, Hamdi Alakkad

This paper presents new approaches to the fixed point property for nonexpansive mappings in L^1 spaces. While it is well-known that L^1 fails the fixed point property in general, we provide a complete and self-contained proof that measure-compactness of a convex set is a sufficient condition. Our exposition makes all compactness and uniform integrability arguments explicit, offering a clear path from measure-theoretic compactness to weak compactness, normal structure, and ultimately fixed points via Kirk's theorem. Beyond this geometric approach, we contextualize this result within broader structural strategies for obtaining fixed points in L^1 and related spaces. We discuss the roles of ultraproducts, equivalent renormings that induce uniform convexity on l^1, and the fixed point property in non-reflexive spaces like Lorentz sequence spaces. This work unifies these perspectives, demonstrating that the obstruction to fixed points in L^1 is not the space itself but specific geometric or structural properties of its subsets. The results clarify the landscape of fixed point theory in non-reflexive Banach spaces.

comment: 10 pages

☆ Random Nonlinear Fusion Frames from Averaged Operator Iterations

James Tian

We study random iterations of averaged operators in Hilbert spaces and prove that the associated residuals converge exponentially fast, both in expectation and almost surely. Our results provide quantitative bounds in terms of a single geometric parameter, giving sharp control of convergence rates under minimal assumptions. As an application, we introduce the concept of random nonlinear fusion frames. Here the atoms are generated dynamically from the residuals of the iteration and yield exact synthesis with frame-like stability in expectation. We show that these frames achieve exponential sampling complexity and encompass important special cases such as random projections and randomized Kaczmarz methods. This reveals a link between stochastic operator theory, frame theory, and randomized algorithms, and establishes a structural tool for constructing nonlinear frame-like systems with strong stability and convergence guarantees.

☆ Amenability, Optimal Transport and Abstract Ergodic Theorems

Christian Rosendal

Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose $d$ is a compatible left-invariant metric on an amenable topological group $G$ with no non-trivial homomorphisms to $\mathbb R$. Then, for every finite subset $E\subseteq G$ and $\epsilon>0$, there is a finitely supported probability measure $\beta$ on $G$ such that $$ \max_{g,h\in E}\, {\sf W}(\beta g, \beta h)<\epsilon, $$ where ${\sf W}$ denotes the Wasserstein distance between probability measures on the metric space $(G,d)$. When $d$ is the word metric on a finitely generated group $G$, this strengthens a well known theorem of Reiter and, when $d$ is bounded, recovers a result of Schneider and Thom. Furthermore, when $G$ is locally compact, $\beta$ may be replaced by an appropriate probability density $f\in L^1(G)$. Also, when $G\curvearrowright X$ is a continuous isometric action on a metric space, the space of Lipschitz functions on the quotient $X/\!\!/G$ is isometrically isomorphic to a $1$-complemented subspace of the Lipschitz functions on $X$. And, when additionally $G$ is skew-amenable, there is a $G$-invariant contraction $$ \mathfrak {Lip}\, X \overset S\longrightarrow\mathfrak{Lip}(X/\!\!/G) $$ so that $(S\phi\big)\big(\overline{Gx}\big)=\phi(x)$ whenever $\phi$ is constant on every orbit of $G\curvearrowright X$. This latter extends results of Cuth and Doucha from the setting of locally compact or balanced groups.

☆ Stability and asymptotic behaviour of one-dimensional solutions in cylinders

Francesca De Marchis, Lisa Mazzuoli, Filomena Pacella

We consider positive one-dimensional solutions of a Lane-Emden relative Dirichlet problem in a cylinder and study their stability/instability properties as the energy varies with respect to domain perturbations. This depends on the exponent $p >1$ of the nonlinearity and we obtain results for $p$ close to 1 and for $p$ large. This is achieved by a careful asymptotic analysis of the one-dimensional solution as $p \to 1$ or $p \to \infty$, which is of independent interest. It allows to detect the limit profile and other qualitative properties of these solutions.

☆ Sharp bilinear eigenfunction estimate, $L^\infty_{x_2}L^p_{t,x_1}$-type Strichartz estimate, and energy-critical NLS

Yangkendi Deng, Yunfeng Zhang, Zehua Zhao

We establish sharp bilinear and multilinear eigenfunction estimates for the Laplace-Beltrami operator on the standard three-sphere $\mathbb{S}^3$, eliminating the logarithmic loss that has persisted in the literature since the pioneering work of Burq, G\'erard, and Tzvetkov over twenty years ago. This completes the theory of multilinear eigenfunction estimates on the standard spheres. Our approach relies on viewing $\mathbb{S}^3$ as the compact Lie group $\mathrm{SU}(2)$ and exploiting its representation theory, especially the properties of Clebsch-Gordan coefficients. Motivated by application to the energy-critical nonlinear Schr\"odinger equation (NLS) on $\mathbb{R} \times \mathbb{S}^3$, we also prove a refined Strichartz estimate of mixed-norm type $L^\infty_{x_2}L^4_{t,x_1}$ on the cylindrical space $\mathbb{R}_{x_1} \times \mathbb{T}_{x_2}$, adapted to certain spectrally localized functions. Combining these two ingredients, we derive a refined bilinear Strichartz estimate on $\mathbb{R} \times \mathbb{S}^3$, which in turn yields small data global well-posedness for the above mentioned NLS in the energy space.

comment: 31 pages. Comments are welcome!

☆ Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates

Andrew Hassell, Qiuye Jia, Ethan Sussman, Andras Vasy

This is the more technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which the equation is approximable by the usual Schrodinger equation. Combining the analyses in the two regimes gives global estimates which are uniform as the speed of light goes to infinity. The companion paper gives applications. Our main technical tools are three new pseudodifferential calculi, $\Psi_{\natural}$ (a variant of the semiclassical scattering calculus), $\Psi_{\natural\mathrm{res}}$, and $\Psi_{\natural2\mathrm{res}}$, the latter two of which are created by ``second microlocalizing'' the first at certain locations. This paper and the companion paper can be read in either order, since the latter treats the former as a black box.

comment: 99 pages, 14 figures

☆ Optimal convergence rates in multiscale elliptic homogenization

Weisheng Niu, Yao Xu, Jinping Zhuge

This paper is devoted to the quantitative homogenization of multiscale elliptic operator $-\nabla\cdot A_\varepsilon \nabla$, where $A_\varepsilon(x) = A(x/\varepsilon_1, x/\varepsilon_2,\cdots, x/\varepsilon_n)$, $\varepsilon = (\varepsilon_1, \varepsilon_2,\cdots, \varepsilon_n) \in (0,1]^n$ and $\varepsilon_i > \varepsilon_{i+1}$. We assume that $A(y_1,y_2,\cdots, y_n)$ is 1-periodic in each $y_i \in \mathbb{R}^d$ and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios $\max \{ \varepsilon_{i+1}/\varepsilon_i: 1\le i\le n-1\}$. In the present paper, under the assumption of real analytic coefficients, we introduce the so-called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to $\max \{ e^{-c\varepsilon_{i}/\varepsilon_{i+1}}: 1\le i\le n-1 \}$. This convergence rate is optimal in the sense that $c>0$ cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double-log scale-separation condition.

comment: 71 pages

☆ New Homogeneous Solutions for the One-Phase Free Boundary Problem

Coleman Hines, James Kolesar, Peter McGrath

For each sufficiently large integer $k$, we construct a domain in the round $2$-sphere with $k$ boundary components which is the link of a cone in $\mathbb{R}^3$ admitting a homogeneous solution to the one-phase free boundary problem. This answers a question of Jerison-Kamburov, and also disproves a conjecture of Souam left open in earlier work. The method exploits a new connection with minimal surfaces, which we also use to construct an infinite family of homogeneous solutions in dimension four.

☆ Well-posedness of stationary 2D and 3D convective Brinkman-Forchheimer extended Darcy Hemivariational inequalities

Manil T. Mohan

This study addresses the well-posedness of a hemivariational inequality derived from the convective Brinkman-Forchheimer extended Darcy (CBFeD) model in both two and three dimensions. The CBFeD model describes the behavior of incompressible viscous fluid flow through a porous medium, incorporating the effects of convection, damping, and nonlinear resistance. The mathematical framework captures steady-state flow conditions under a no-slip boundary assumption, with a non-monotone boundary condition that links the total fluid pressure and the velocity's normal component through a Clarke subdifferential formulation. To facilitate the analysis, we introduce an auxiliary hemivariational inequality resembling a nonlinear Stokes-type problem with damping and pumping terms, which serves as a foundational tool in establishing the existence and uniqueness of weak solutions for the CBFeD model. The analytical strategy integrates techniques from convex minimization theory with fixed-point methods, specifically employing either the Banach contraction mapping principle or Schauder's fixed point theorem. The Banach-based approach, in particular, leads to a practical iterative algorithm that solves the original nonlinear hemivariational inequality by sequentially solving Stokes-type problems, ensuring convergence of the solution sequence. Additionally, we derive equivalent variational formulations in terms of minimization problems. These formulations lay the groundwork for the design of efficient and stable numerical schemes tailored to simulate flows governed by the CBFeD model.

☆ Numerical analysis of the homogeneous Landau equation: approximation, error estimates and simulation

Francis Filbet, Yanzhi Gui, Ling-Bing He

We construct a numerical solution to the spatially homogeneous Landau equation with Coulomb potential on a domain $D_L$ with N retained Fourier modes. By deriving an explicit error estimate in terms of $L$ and $N$, we demonstrate that for any prescribed error tolerance and fixed time interval $[0, T ]$, there exist choices of $D_L$ and $N$ satisfying explicit conditions such that the error between the numerical and exact solutions is below the tolerance. Specifically, the estimate shows that sufficiently large $L$ and $N$ (depending on initial data parameters and $T$) can reduce the error to any desired level. Numerical simulations based on this construction are also presented. The results in particular demonstrate the mathematical validity of the spectral method proposed in the referenced literature.

☆ The role of communication delays in the optimal control of spatially invariant systems

Luca Ballotta, Juncal Arbelaiz, Vijay Gupta, Luca Schenato, Mihailo R. Jovanović

We study optimal proportional feedback controllers for spatially invariant systems when the controller has access to delayed state measurements received from different spatial locations. We analyze how delays affect the spatial locality of the optimal feedback gain leveraging the problem decoupling in the spatial frequency domain. For the cases of expensive control and small delay, we provide exact expressions of the optimal controllers in the limit for infinite control weight and vanishing delay, respectively. In the expensive control regime, the optimal feedback control law decomposes into a delay-aware filtering of the delayed state and the optimal controller in the delay-free setting. Under small delays, the optimal controller is a perturbation of the delay-free one which depends linearly on the delay. We illustrate our analytical findings with a reaction-diffusion process over the real line and a multi-agent system coupled through circulant matrices, showing that delays reduce the effectiveness of optimal feedback control and may require each subsystem within a distributed implementation to communicate with farther-away locations.

comment: {\copyright} 2025 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

☆ Functions of bounded Musielak-Orlicz-type deformation and anisotropic Total Generalized Variation for image-denoising problems

Giacomo Bertazzoni, Elisa Davoli, Samuele Ricco`, Elvira Zappale

In the first part of this paper we introduce the space of bounded deformation fields with generalized Orlicz growth. We establish their main properties, provide a modular representation, and characterize a decomposition of the modular into an absolutely continuous part and a singular part weighted via a recession function. A further analysis in the variable exponent case is also provided. The second part of the paper contains a notion of Musielak-Orlicz anisotropic Total Generalized Variation. We establish a duality representation, and show well-posedness of the corresponding image reconstruction problem.

☆ On the Convergence of Solutions for the Ginzburg-Landau Equation and System

Rejeb Hadiji, Jongmin Han

Let $(u_\varepsilon)$ be a family of solutions of the Ginzburg--Landau equation with boundary condition $u_\varepsilon = g$ on $\partial \Omega$ and of degree $0$. Let $u_0$ denote the harmonic map satisfying $u_0 = g$ on $\partial \Omega$. We show that, if there exists a constant $C_1 > 0$ such that for $\varepsilon$ sufficiently small we have $\frac{1}{2} \int_\Omega |\nabla u_\ve|^2 dx \leq C_1 \leq \frac{1}{2} \int_\Omega |\nabla u_0|^2 dx,$ then $C_1 = \frac{1}{2} \int_\Omega |\nabla u_0|^2 dx$ and $u_\ve ~\to ~ u_0 \qin H^1(\Om)$. We also prove that if there is a constant $C_2$ such that for $\ve$ small enough we have $ \frac12 \int_\Om |\nabla u_\ve|^2 dx \geq C_2 > \frac12 \int_\Om |\nabla u_0|^2 dx,$ then $|u_{\ve}|$ does not converge uniformly to $1$ on $\overline{\Om} $. We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg--Landau systems.

☆ Mountain Pass Critical Points of the Liquid Drop Model

Gregory R. Chambers, Jared Marx-Kuo

We consider Gamow's liquid drop functional, $\mathcal{E}$, on $\mathbb{R}^3$ and construct non-minimizing, volume constrained, critical points for volumes $3.512 \cong \alpha_0 < V < 10$. In this range, we establish a mountain pass set up between a ball of volume $V$ and two balls of volume $V/2$ infinitely far apart. Intuitively, our critical point corresponds to the maximal energy configuration of an atom of volume $V$ as it undergoes fission into two atoms of volume $V/2$. Our proof relies on geometric measure theoretical methods from the min-max construction of minimal surfaces, and along the way, we address issues of non-compactness, ``pull tight" with a volume constraint, and multiplicity.

comment: 17 pages, 1 figure, comments welcome!

☆ Homogenization of rate-independent elastoplastic spring network models with non-local random fields

Simone Hermann

We investigate the time-evolution of elastoplastic materials reinforced by randomly distributed long-range interactions. Starting from a rate-independent system on a discrete spring lattice that combines local linearized elasticity, gradient-regularized plasticity and stochastic non-local links modeling stiff fibers, we establish a discrete-to-continuum limit in the energetic formulation. We prove that as the lattice spacing tends to zero, an evolutionary solution of the discrete system converges to the unique energetic solution of a continuum limit problem. The limiting continuum model couples classical elastoplasticity with a non-local energy featuring fractional-order interactions that capture the homogenized influence of random long-range reinforcements. These results extend previous static homogenization studies by rigorously treating path-dependent dissipation and showing existence, uniqueness and Lipschitz continuity of the evolving solutions. The work therefore provides a mathematical foundation for simulating time-dependent mechanical response of fiber-reinforced composites with random architecture.

☆ Strong and weak solutions to a structural acoustic model with a $C^1$ source term on the plate

Andrew R. Becklin, Yanqiu Guo

In this manuscript, we consider a structural acoustic model consisting of a wave equation defined in a bounded domain $\Omega \subset \mathbb{R}^3$, strongly coupled with a Berger plate equation acting on the flat portion of the boundary of $\Omega$. The system is influenced by an arbitrary $C^1$ nonlinear source term in the plate equation. Using nonlinear semigroup theory and monotone operator theory, we establish the well-posedness of both local strong and weak solutions, along with conditions for global existence. With additional assumptions on the source term, we examine the Nehari manifold and establish the global existence of potential well solutions. Our primary objective is to characterize regimes in which the system remains globally well-posed despite arbitrary growth of the source term and the absence of damping mechanisms to stabilize the dynamics.

☆ Local uniqueness and non-degeneracy of blowup solutions for regular Liouville systems

Zetao Cheng, Haoyu Li, Lei Zhang

We study the following Liouville system defined on a compact Riemann surface $M$, \begin{equation} -\Delta u_i=\sum_{j=1}^n a_{ij}\rho_j\Big(\frac{h_j e^{u_j}}{\int_\Omega h_j e^{u_j}}-1\Big)\mbox{ in }M\mbox{ for }i=1,\cdots,n,\nonumber \end{equation} where the coefficient matrix $A=(a_{ij})_{n\times n}$ is nonnegative, $h_1, \ldots, h_n$ are positive smooth functions, and $\rho_1, \ldots, \rho_n$ are positive constants. For the blowup solutions, we establish their uniqueness and non-degeneracy based on natural assumptions. The main results significantly generalize corresponding results for single Liouville equations \cite{BartJevLeeYang2019,BartYangZhang20241,BartYangZhang20242}. To overcome several substantial difficulties, we develop certain tools and extend them into a more general framework applicable to similar situations. Notably, to address the considerable challenge of a continuum of standard bubbles, we refine the techniques from Huang-Zhang \cite{HuangZhang2022} and Zhang \cite{Zhang2006,Zhang2009} to achieve extremely precise pointwise estimates. Additionally, to address the limited information provided by the Pohozaev identity, we develop a useful Fredholm theory to discern the exact role that the Pohozaev identity plays for systems. The considerable difference between systems and a single equation is also reflected in the location of blowup points, where the uncertainty of the energy type of the blowup point makes it difficult to determine the sufficiency of pointwise estimates. In this regard, we extend our highly precise pointwise estimates to any finite order. This aspect is drastically distinct from analyses of single equations.

comment: 53 pages, one figure

☆ $H^\infty$-calculus for the Dirichlet Laplacian on angular domains

Petru A. Cioica-Licht, Emiel Lorist, P. Tobias Werner

We establish boundedness of the $H^\infty$-calculus for the Dirichlet Laplacian on angular domains in $\R^2$ and corresponding wedges on $L^p$-spaces with mixed weights. The weights are based on both the distance to the boundary and the distance to the tip of the underlying domain. Our main motivation comes from the study of stochastic partial differential equations and associated degenerate deterministic parabolic equations on non-smooth domains. As a consequence of our analysis, we also obtain maximal $L^p$-regularity for the Poisson equation on angular domains in appropriate weighted Sobolev spaces.

comment: 30 pages

♻ ☆ Beyond separability: convergence rate of vanishing viscosity approximations to mean field games via FBSDE stability

Winston Yu, Qiang Du, Wenpin Tang

This paper studies the vanishing viscosity approximation to mean field games (MFGs) in $\mathbb{R}^d$ with a nonlocal and possibly non-separable Hamiltonian. We prove that the value function converges at a rate of $\mathcal{O}(\beta)$, where $\beta^2$ is the diffusivity constant, which matches the classical convergence rate of vanishing viscosity for Hamilton-Jacobi (HJ) equations. The same rate is also obtained for the approximation of the distribution of players as well as for the gradient of the value function. The proof is a combination of probabilistic and analytical arguments by first analyzing the forward-backward stochastic differential equation associated with the MFG, and then applying a general stability result for HJ equations. Applications of our result to $N$-player games, mean field control, and policy iteration for solving MFGs are also presented.

comment: 25 pages, 2 figures

♻ ☆ Asymptotic stability of the composite wave of rarefaction wave and contact wave to nonlinear viscoelasticity model with non-convex flux

Zhenhua Guo, Meichen Hou, Guiqin Qiu, Lingda Xu

In this paper, we consider the wave propagations of viscoelastic materials, which has been derived by Taiping-Liu to approximate the viscoelastic dynamic system with fading memory (see [T.P.Liu(1988)\cite{LiuTP}]) by the Chapman-Enskog expansion. By constructing a set of linear diffusion waves coupled with the high-order diffusion waves to achieve cancellations to approximate the viscous contact wave well and explicit expressions, the nonlinear stability of the composite wave is obtained by a continuum argument. It emphasis that, the stress function in our paper is a general non-convex function, which leads to several essential differences from strictly hyperbolic systems such as the Euler system. Our method is completely new and can be applied to more general systems and a new weighted Poincar\'e type of inequality is established, which is more challenging compared to the convex case and this inequality plays an important role in studying systems with non-convex flux.

♻ ☆ An approximate solution of a case of perturbed Fokker-Planck equation

Yan Luo, Kaicheng Sheng

This paper focuses on finding an approximate solution of a kind of Fokker-Planck equation with time-dependent perturbations. A formulation of the approximate solution of the equation is constructed, and then the existence of the formulation is proved. The related Hamiltonian dynamical system explains the estimations. Our work provides a more comprehensive understanding of the behaviour of systems described by this Fokker-Planck equation and the corresponding stochastic differential equation.

♻ ☆ The Christoffel problem for the disk area measure

Leo Brauner, Georg C. Hofstätter, Oscar Ortega-Moreno

The mixed Christoffel problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, all but one of them are fixed. We provide a solution in the case where the fixed reference bodies are (n-1)-dimensional disks with parallel axes and the measure has no mass at the poles of the sphere determined by this axis.

comment: 9 pages

♻ ☆ Mixed Christoffel-Minkowski problems for bodies of revolution

Leo Brauner, Georg C. Hofstätter, Oscar Ortega-Moreno

The mixed Christoffel-Minkowski problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, one of which, appearing multiple times, is free and the rest are fixed. In the case where all bodies involved are symmetric around a common axis, we provide a complete solution to this problem, without assuming any regularity. In particular, we refine Firey's classification of area measures of figures of revolution. In our argument, we introduce an easy way to transform mixed area measures and mixed volumes involving axially symmetric bodies, and we significantly improve Firey's estimate on the local behavior of area measures. As a secondary result, we obtain a family of Hadwiger type theorems for convex valuations that are invariant under rotations around an axis.

comment: 45 pages

♻ ☆ Propagation of Love waves in linear elastic isotropic Cosserat materials

Marius Apetrii, Emilian Bulgariu, Ionel-Dumitrel Ghiba, Hassam Khan, Patrizio Neff

We investigate the propagation of Love waves in an isotropic half-space modelled as a linear {elastic isotropic} Cosserat material. To this aim, we show that a method commonly used to study Rayleigh wave propagation is also applicable to the analysis of Love wave propagation. This approach is based on the explicit solution of an algebraic Riccati equation, which operates independently of the traditional Stroh formalism. The method provides a straightforward numerical algorithm to determine the wave amplitudes and speed{s}. Beyond its numerical simplicity, the method guarantees the existence and uniqueness of a subsonic wave speed, addressing a problem that remains unresolved in most Cosserat solids generalised {continua} theories. Although often overlooked, proving the existence of an admissible solution is, in fact, the key point that validates or invalidates the entire analytical approach used to derive the equation determining the wave speed. Interestingly, it is confirmed that the Love waves do not need the artificial introduction of a surface layer, as indicated in the literature.

comment: arXiv admin note: substantial text overlap with arXiv:2104.13143

♻ ☆ On the semi-additivity of the $1/2$-symmetric caloric capacity

Joan Hernández, Joan Mateu, Laura Prat

In this paper we study properties of a variant of the $1/2$-caloric capacity, called $1/2$-symmetric caloric capacity. The latter is associated simultaneously with the $1/2$-fractional heat equation and its conjugate. We establish its semi-additivity in $\mathbb{R}^{n+1}$ and, moreover, we compute explicitly the $1/2$-symmetric caloric capacity of rectangles, which illustrates its anisotropic behavior.

♻ ☆ Blow-up for a Nonlocal Diffusion Equation with Time Regularly Varying Nonlinearity and Forcing

Rihab Ben Belgacem, Mohamed Majdoub

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho \mathbf{w}(x), \qquad (x,t)\in \mathbb{R}^N\times (0,\infty), \] where $\mathsf{s}\in (0,1)$, $p>1$, $b\geq 0$, and $\varrho>-1$. The function $\mathsf{h}(t)$ is assumed to belong to the generalized class of regularly varying functions, while $\mathbf{w}$ is a prescribed spatial source. We first revisit the unforced case and establish sharp blow-up and global existence criteria in terms of the critical Fujita exponent, thereby extending earlier results to the wider class of time-dependent coefficients. For the forced problem, we derive nonexistence of global weak solutions under suitable growth conditions on $\mathsf{h}$ and integrability assumptions on $\mathbf{w}$. Furthermore, we provide sufficient smallness conditions on the initial data and the forcing term ensuring global-in-time mild solutions. Our analysis combines semigroup estimates for the mixed operator, test function methods, and asymptotic properties of regularly varying functions. To our knowledge, this is the first study addressing blow-up phenomena for nonlinear diffusion equations with such a class of time-dependent coefficients.

comment: 21 pages. Typos have been corrected

♻ ☆ Optimal decay rates to the contact wave for 1-D compressible Navier-Stokes equations

Lingjun Liu, Shu Wang, Lingda Xu

This paper investigates the decay rates of the contact wave in one-dimensional Navier-Stokes equations. We study two cases of perturbations, with and without zero mass condition, i.e., the integration of initial perturbations is zero and non-zero, respectively. For the case without zero mass condition, we obtain the optimal decay rate $(1+t)^{-\frac{1}{2}}$ for the perturbation in $L^\infty$ norm, which provides a positive answer to the conjecture in \cite{HMX}. We applied the anti-derivative method, introducing the diffusion wave to carry the initial excess mass, diagonalizing the integrated system, and estimating the energy of perturbation in the diagonalized system. Precisely, due to the presence of diffusion waves, the decay rates for errors of perturbed system are too poor to get the optimal decay rate. We find the dissipation structural in the diagonalized system, see \cref{ds}. This observation makes us able to fully utilize the fact that the sign of the derivative of the contact wave is invariant and to control the terms with poor decay rates in energy estimates. For the case with zero mass condition, there are also terms with poor decay rates. In this case, note that there is a cancellation in the linearly degenerate field so that the terms with poor decay rates will not appear in the second equation of the diagonalized system. Thanks to this cancellation and a Poincar\'e type of estimate obtained by a critical inequality introduced by \cite{HLM}, we get the decay rate of $\ln^{\frac{1}{2}} (2+t)$ for $L^2$ norm of anti-derivatives of perturbation and $(1+t)^{-\frac{1}{2}}\ln^{\frac{1}{2}}(2+t)$ for the $L^2$ norm of perturbation itself, the decay rates are optimal, which is consistent with the results obtained by using pointwise estimate in \cite{XZ} for the system with artificial viscosity.

♻ ☆ Localization and global dynamics in the long-range discrete nonlinear Schrödinger equation

Brian Choi, Austin Marstaller, Alejandro Aceves

We study localization, pinning, and mobility in the fractional discrete nonlinear Schr\"odinger equation (fDNLS) with generalized power-law coupling. A finite-dimensional spatial-dynamics reduction of the nonlocal recurrence yields onsite and offsite stationary profiles; their asymptotic validity, orbital stability of onsite solutions, and $\ell^2$ proximity to the exact lattice solutions are established. Using the explicit construction of localized states, it is shown that the spatial tail behavior is algebraic for all $\alpha$ > 0. The Peierls-Nabarro barrier (PNB) is computed, and the parameter regimes are identified where it nearly vanishes; complementary numerical simulations explore mobility/pinning across parameters and exhibit scenarios consistent with near-vanishing PNB. We also analyze modulational instability of plane waves, locate instability thresholds, and discuss the role of nonlocality in initiating localization. Finally, we establish small-data scattering, and quantify how fDNLS dynamics approximates the nearest-neighbor DNLS on bounded times while exhibiting distinct global behavior for any large $\alpha$.

comment: Please update the secondary classification: remove math.AP (Analysis of PDEs) and add math.DS (Dynamical Systems). The primary category should remain unchanged

♻ ☆ A qualitative study of the generalized dispersive systems with time-delay: The unbounded case

Roberto de A. Capistrano Filho, Fernando Gallego, Vilmos Komornik

We study the asymptotic behavior of the solutions of the time-delayed higher-order dispersive nonlinear differential equation \begin{equation*} u_t(x,t)+Au(x,t) +\lambda_0(x) u(x,t)+\lambda(x) u(x,t-\tau )=0 \end{equation*} where \begin{equation*} Au=(-1)^{j+1}\partial_x^{2j+1}u+(-1)^m\partial_x^{2m}u+ \frac{1}{p+1}\partial_xu^{p+1} \end{equation*} with $m\le j$ and $1\le p<2j$. Under suitable assumptions on the time delay coefficients, we prove that the system is exponentially stable if the coefficient of the delay term is bounded from below by a suitable positive constant, without any assumption on the sign of the coefficient of the undelayed feedback. Additionally, in the absence of delay, general results of stabilization are established in $H^s(\mathbb{R})$ for $s\in[0,2j+1]$. Our results generalize several previous theorems for the Korteweg-de Vries type delayed systems in the literature.

comment: To appear on Journal of Evolution Equations

☆ Uniqueness of Hahn--Banach extensions and inner ideals in real C$^*$-algebras and real JB$^*$-triples

Lei Li, Antonio M. Peralta, Shanshan Su, Jiayin Zhang

We show that every closed (resp., weak$^*$-closed) inner ideal $I$ of a real JB$^*$-triple (resp. a real JBW$^*$-triple) $E$ is Hahn--Banach smooth (resp., weak$^*$-Hahn--Banach smooth). Contrary to what is known for complex JB$^*$-triples, being (weak$^*$-)Hahn--Banach smooth does not characterise (weak$^*$-)closed inner ideals in real JB(W)$^*$-triples. We prove here that a closed (resp., weak$^*$-closed) subtriple of a real JB$^*$-triple (resp., a real JBW$^*$-triple) is Hahn-Banach smooth (resp., weak$^*$-Hahn-Banach smooth) if, and only if, it is a hereditary subtriple. If we assume that $E$ is a reduced and atomic JBW$^*$-triple, every weak$^*$-closed subtriple of $E$ which is also weak$^*$-Hahn-Banach smooth is an inner ideal.\smallskip In case that $C$ is the realification of a complex Cartan factor or a non-reduced real Cartan factor, we show that every weak$^*$-closed subtriple of $C$ which is weak$^*$-Hahn-Banach smooth and has rank $\geq 2$ is an inner ideal. The previous conclusions are finally combined to prove the following: Let $I$ be a closed subtriple of a real JB$^*$-triple $E$ satisfying the following hypotheses: $(a)$ $I^*$ is separable. $(b)$ $I$ is weak$^*$-Hahn-Banach smooth. $(c)$ The projection of $I^{**}$ onto each real or complex Cartan factor summand in the atomic part of $E^{**}$ is zero or has rank $\geq 2$. Then $I$ is an inner ideal of $E$.

☆ $H^\infty$-calculus for the Dirichlet Laplacian on angular domains

Petru A. Cioica-Licht, Emiel Lorist, P. Tobias Werner

We establish boundedness of the $H^\infty$-calculus for the Dirichlet Laplacian on angular domains in $\R^2$ and corresponding wedges on $L^p$-spaces with mixed weights. The weights are based on both the distance to the boundary and the distance to the tip of the underlying domain. Our main motivation comes from the study of stochastic partial differential equations and associated degenerate deterministic parabolic equations on non-smooth domains. As a consequence of our analysis, we also obtain maximal $L^p$-regularity for the Poisson equation on angular domains in appropriate weighted Sobolev spaces.

comment: 30 pages

☆ A note concerning fundamental functions of interpolation associated to the inverse multiquadric $(α^2+x^2)^{-k}$

Jeff Ledford, Kyle Rutherford

This short note develops fundamental functions associated with the scattered shifts of the inverse \emph{multiquadric} function $(\alpha^2 + x^2)^{-k}$, for $k\in\mathbb{N}$.

comment: 7 pages

☆ Approximation by Neural Network operators in $L^p$ spaces associated with an arbitrary measure

Nitin Bartwal, A. Sathish Kumar

In this paper, we investigate the approximation behavior of both one and multidimensional neural network type operators for functions in $L^p(I^d,\rho)$, where $1\leq p<\infty$, associated with a general measure $\rho$ defined over a hypercube. First, we prove the uniform approximation for a continuous function and the $L^p$ approximation theorem by the NN operators in one and multidimensional settings. In addition, we also obtain the $L^p$ error bounds in terms of $\mathcal{K}$-functionals for these neural network operators. Finally, we consider the logistic and tangent hyperbolic activation functions and verify the hypothesis of the theorems. We also show the implementation of continuous and integrable functions by NN operators with respect to the Lebesgue and Jacobi measures defined on $[0,1]\times[0,1]$ with logistic and tangent hyperbolic activation functions.

☆ Inequality, uncertainty principles and their structural analysis for offset linear canonical transform and its quaternion extension

Gita Rani Mahato, Sarga Varghese, Manab Kundu

This work undertakes a twofold investigation. In the first part, we examine the inequalities and uncertainty principles in the framework of offset linear canonical transform (OLCT), with particular attention to its scaling and shifting effects. Theoretical developments are complemented by numerical simulations that substantiate and illustrate the analytical results. In the second part, we establish the connection of quaternion offset linear canonical transform (QOLCT) and the OLCT by employing the orthogonal plane split (OPS) approach. Through this approach, the inequalities and uncertainty principles derived for the OLCT are extended to the QOLCT. Moreover, the computational methods designed for the OLCT may be systematically adapted to facilitate the numerical implementation of the QOLCT using this connection between OLCT and QOLCT.

☆ Index theory for non-compact quantum graphs

Daniele Garrisi, Alessandro Portaluri, Li Wu

We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating the negative directions of the Lagrangian action with the total multiplicity of conjugate instants along the edges. These results extend classical tools in global analysis and symplectic geometry to graph based models, with applications to nonlinear wave equations such as the nonlinear Schroedinger equation. The spectral flow formula is proved by constructing a Lagrangian intersection theory in the Gelfand-Robbin quotients of the second variation of the action. This approach also recovers, in a unified way, the known formulas for heteroclinic, halfclinic, homoclinic, and bounded orbits of (non)autonomous Lagrangian systems.

comment: 35 pages, 3 figures

☆ The Structure of Extremal Bad Science Matrices

Shridhar Sinha

We study the 'bad science matrix problem': among all matrices $A\in\mathbb{R}^{n\times n}$ whose rows have unit $\ell_2$-norm, determine the maximum of $\beta(A)=\frac{1}{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty$. Steinerberger [1] (arXiv:2402.03205) showed that the optimal asymptotic rate is $(1+o(1))\sqrt{2\log n}$, and that this rate is attained with high probability by matrices with i.i.d. $\pm1$ entries after normalization. More recent explicit constructions [2] (arXiv:2408.00933) achieve $\beta(A)\ge\sqrt{\log_2(n)+1}$, which lies within a constant factor of the asymptotic optimum. In this paper we bridge the gap between the probabilistic and explicit approaches. We give a geometric description of extremizers as (nearly) isoperimetrically extremal partitions of the $n$-dimensional hypercube induced by the rows of $A$. We obtain precise rates for heuristic constructions by recasting the maximization of $\beta(A)$ in the language of high-dimensional central-limit theorems as in Fang, Koike, Liu and Zhao [16] (arXiv:2305.17365). Using these connections, we present a family of explicit deterministic matrices $A_n$ that exist for all $n$ under the assumption of Hadamard's conjecture, and for infinitely many $n$ unconditionally, such that for all $n$ sufficiently large $\beta(A_n)\ge\bigl(1 - \frac{\log\log(2n)}{4\log(2n)}\bigr)\sqrt{2\log(2n)}.$

comment: 27 pages, 6 figures

♻ ☆ A Rockafellar Theorem for cyclically quasi-monotone maps: the regular non-vanishing case

Luigi De Pascale, Paul Pegon

We study the connection between cyclic quasi-monotonicity and quasi-convexity, focusing on whether every cyclically quasi-monotone (possibly multivalued) map is included in the normal cone operator of a quasi-convex function, in analogy with Rockafellar's theorem for convex functions. We provide a positive answer for $\mathscr{C}^1$-regular, non-vanishing maps in any dimension, as well as for general multi-maps in dimension $1$. We further discuss connections to revealed preference theory in economics and to $L^\infty$ optimal transport. Finally, we present explicit constructions and examples, highlighting the main challenges that arise in the general case.

comment: 33 pages, 7 figures

♻ ☆ Quantization dimensions of negative order

Marc Kesseböhmer, Aljoscha Niemann

We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order $r$, including negative values of $r$. To this end, we use the concept of partition functions, which generalizes the idea of the $L^{q}$-spectrum and in this way naturally extends the work in [M. Kesseb\"ohmer, A. Niemann, and S. Zhu. Quantization dimensions of probability measures via R\'enyi dimensions. Trans. Amer. Math. Soc. 376.7 (2023)]. In particular, we provide natural fractal geometric bounds as well as easily verifiable necessary conditions for the existence of the quantization dimensions. The exact asymptotics of the quantization error of negative order for absolutely continuous measures are stated, whereby an open question from [S. Graf, H. Luschgy. Math. Proc. Cambridge Philos. Soc. 136, 3 (2004)] regarding the geometric mean error is also answered in the affirmative.

comment: 19 pages, 1 figure

♻ ☆ On the limiting distribution of sums of random multiplicative functions

Ofir Gorodetsky, Mo Dick Wong

We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved by applying the martingale central limit theorem to a suitably truncated sum. This truncation is inspired by work of Najnudel, Paquette, Simm and Vu on subcritical holomorphic multiplicative chaos setting, but analysed with a different conditioning argument generalised from Harper's work on fractional moments to circumvent integrability issues at criticality. A significant part of the proof is devoted to the convergence in probability of the associated partial Euler product to a critical multiplicative chaos measure, independent of the mild shift away from the critical line. Our approach to the universality of critical non-Gaussian multiplicative chaos bypasses the barrier analysis with the help of a modified second moment method, and employs a novel argument based on coupling and homogenisation by change of measure, which could be of independent interest.

comment: 46 pages; some minor corrections and added references

♻ ☆ Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian

Jesús Núñez-Zimbrón, Enrico Pasqualetto, Elefterios Soultanis

We show that a metric space $X$ that, at every point, has a Gromov-Hausdorff tangent with the splitting property (i.e. every geodesic line splits off a factor $\mathbb{R}$), is universally infinitesimally Hilbertian (i.e. $W^{1,2}(X,\mu)$ is a Hilbert space for every measure $\mu$). This connects the infinitesimal geometry of $X$ to its analytic properties and is, to our knowledge, the first general criterion guaranteeing universal infinitesimal Hilbertianity. Using it we establish universal infinitesimal Hilbertianity of finite dimensional RCD-spaces. We moreover show that (possibly infinite dimensional) Alexandrov spaces are universally infinitesimally Hilbertian and construct an isometric embedding of tangent modules.

comment: 19 pages

♻ ☆ Nonlocal BV and nonlocal Sobolev spaces induced by nonfractional weight functions

Francesc Alcover, Joan Duran, Ramon Oliver-Bonafoux, Catalina Sbert

In this paper, we expand upon the theory of the space of functions with nonlocal weighted bounded variation, first introduced by Kindermann et.al. in 2005 and later generalized by Wang et.al. in 2014. We consider nonfractional C^1 weights and, using an analogous formulation to the aforementioned works, we also introduce a (to our knowledge) new class of nonlocal weighted Sobolev spaces. After establishing some fundamental properties and results regarding the structure of these spaces, we study their relationship with the classical BV and Sobolev spaces, as well as with the space of test functions. We handle both the case of domains with finite measure and that of domains of infinite measure, and show that these two situations lead to quite different scenarios. As an application, we also show that these function spaces are suitable for establishing existence and uniqueness results of global minimizers for several classes of functionals. Some of these functionals were introduced in the above-mentioned references for the study of image deblurring problems.

♻ ☆ Convexity and concavity of $f$-potentials (Kolmogorov means)

V. I. Bakhtin, N. A. Tsarev

In the paper we prove criteria for convexity and concavity of $f$-potentials ($f$-means, Kolmogorov means, weighted quasi-arithmetic means), which particular cases are the arithmetic, geometric, harmonic means, the thermodynamic potential (exponential mean), and the $L^{p}$-norm. Then we compute in quadratures all functions $f$ satisfying these criteria.

comment: 13 pages, some references and comments about priority added

♻ ☆ A Note on Carlier Inequality

Regina S. Burachik, J. E. Martínez-Legaz

Recently, Carlier established in [3] a quantitave version of the Fitzpatrick inequality in a Hilbert space. We extend this result by Carlier to the framework of reflexive Banach spaces. In the Hilbert space setting, we obtain an improved version of the strong Fitzpatrick inequality due to Voisei and Z\u{a}linescu.

comment: 7 pages

☆ Sharp power concavity of two relevant free boundary problems of reaction-diffusion type

Qingyou He

The porous medium type reaction-diffusion equation and the Hele-Shaw problem are two free boundary problems linked through the incompressible (Hele-Shaw) limit. We investigate and compare the sharp power concavities of the pressures on their respective supports for the two free boundary problems. For the pressure of the porous medium type reaction-diffusion equation, the $\frac{1}{2}$-concavity preserves all the time, while $\alpha$-concavity for $\alpha\in[0,\frac{1}{2})\cup(\frac{1}{2},1]$ does not persist in time. In contrast, in the case of the pressure for the Hele-Shaw problem, $\alpha$-concavity with $\alpha\in[0,\frac{1}{2}]$ is maintained all the while and $\frac{1}{2}$ acts as the largest index. The intuitive explanation for the difference between the two free boundary problems is that, although the Hele-Shaw problem is the incompressible limit of the porous medium-type reaction-diffusion equation, it is no longer a degenerate parabolic equation. Furthermore, for the pressure of the porous medium type reaction-diffusion equation, the non-degenerate estimate is established by means of the derived concave properties, indicating that the spatial Lipschitz regularity in the whole space is sharp.

☆ Existence of minimizers for interaction energies with external potentials

Ruiwen Shu

In this paper we study the existence of minimizers for interaction energies with the presence of external potentials. We consider a class of subharmonic interaction potentials, which include the Riesz potentials $|{\bf x}|^{-s},\,\max\{0,d-2\}

☆ Lipschitz regularity for $p$-harmonic interface transmission problems

Marius Müller

We prove optimal Lipschitz regularity for weak solutions of the measure-valued $p$-Poisson equation $-\Delta_p u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$. Here $p \in (1,2)$, $\Gamma$ is a compact and connected $C^2$-hypersurface without boundary, and $Q$ is a positive $W^{2,\infty}$-density. This equation can be understood as a nonlinear interface transmission problem. Our main result extends previous studies of the linear case and provides further insights on a delicate limit case of (linear and nonlinear) potential theory.

comment: 18 pages, comments welcome!

☆ On the dichotomy of $p$-walk dimensions on metric measure spaces

Meng Yang

On a volume doubling metric measure space endowed with a family of $p$-energies such that the Poincar\'e inequality and the cutoff Sobolev inequality with $p$-walk dimension $\beta_p$ hold, for $p$ in an open interval $I\subseteq (1,+\infty)$, we prove the following dichotomy: either $\beta_p=p$ for all $p\in I$, or $\beta_p>p$ for all $p\in I$.

comment: 21 pages

☆ Uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem

Yao Wan

This paper investigates the spectral properties of the Hilbert-Brunn-Minkowski operator $L_K$ to derive stability estimates for geometric inequalities, including the local Brunn-Minkowski inequality. By analyzing the eigenvalues of $L_K$, we establish the uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem in $\mathbb{R}^{n}$ for $\frac{1-3n^2}{2n}\leq p<-n$ with $\lambda_2(-L_K)\geq \frac{n-1}{2n-1+p}$. Furthermore, we extend this uniqueness result to the range $-2n-1 \leq p<-n$ with $\lambda_2(-L_K)\geq \frac{-p-1}{n-1}$, assuming the origin-centred condition.

comment: 21 pages. All comments are welcome

☆ A transport approach to the cutoff phenomenon

Francesco Pedrotti, Justin Salez

Substantial progress has recently been made in the understanding of the cutoff phenomenon for Markov processes, using an information-theoretic statistics known as varentropy [Sal23; Sal24; Sal25a; PS25]. In the present paper, we propose an alternative approach which bypasses the use of varentropy and exploits instead a new W-TV transport inequality, combined with a classical parabolic regularization estimate [BGL01; OV01]. While currently restricted to non-negatively curved processes on smooth spaces, our argument no longer requires the chain rule, nor any approximate version thereof. As applications, we recover the main result of [Sal25a] establishing cutoff for the log-concave Langevin dynamics, and extend the conclusion to a widely-used discrete-time sampling algorithm known as the Proximal Sampler.

comment: 11 pages

☆ Linear Convergence of Gradient Descent for Quadratically Regularized Optimal Transport

Alberto González-Sanz, Marcel Nutz, Andrés Riveros Valdevenito

In optimal transport, quadratic regularization is an alternative to entropic regularization when sparse couplings or small regularization parameters are desired. Here quadratic regularization means that transport couplings are penalized by the squared $L^2$ norm, or equivalently the $\chi^2$ divergence. While a number of computational approaches have been shown to work in practice, quadratic regularization is analytically less tractable than entropic, and we are not aware of a previous theoretical convergence rate analysis. We focus on the gradient descent algorithm for the dual transport problem in continuous and semi-discrete settings. This problem is convex but not strongly convex; its solutions are the potential functions that approximate the Kantorovich potentials of unregularized optimal transport. The gradient descent steps are straightforward to implement, and stable for small regularization parameter -- in contrast to Sinkhorn's algorithm in the entropic setting. Our main result is that gradient descent converges linearly; that is, the $L^2$ distance between the iterates and the limiting potentials decreases exponentially fast. Our analysis centers on the linearization of the gradient descent operator at the optimum and uses functional-analytic arguments to bound its spectrum. These techniques seem to be novel in this area and are substantially different from the approaches familiar in entropic optimal transport.

☆ Maximal regularity of Dirichlet problem for the Laplacian in Lipschitz domains

Chérif Amrouche, Mohand Moussaoui

The focus of this work is on the homogeneous and non-homogeneous Dirichlet problem for the Laplacian in bounded Lipschitz domains (BLD). Although it has been extensively studied by many authors, we would like to return to a number of fundamental questions and known results, such as the traces and the maximal regularity of solutions. First, to treat non-homogeneous boundary conditions, we rigorously define the notion of traces for non regular functions. This approach replaces the non-tangential trace notion that has dominated the literature since the 1980s. We identify a functional space E = \{v\in H^{1/2}(\Omega);\nabla v\in [H^1/2(\Omega)]'\} for which the trace operator is continuous from $E$ into $L^2(\Gamma)$. Second, we address the regularity of solutions to the Laplace equation with homogeneous Dirichlet conditions. Using specific equivalent norms in fractional Sobolev spaces and Grisvard's results for polygons and polyhedral domains, we prove that maximal regularity $H^{3/2}$ holds in any BLD $\Omega$, for all right-hand sides in the dual of $H^{1/2}_{00}(\Omega)$. This conclusion contradicts the prevailing claims in the literature since the 1990s. Third, we describe some criteria which establish new uniqueness results for harmonic functions in Lipschitz domains. In particular, we show that if $u\in H^{1/2}(\Omega)$ or $u\in W^{1, 2N/(N+1)}(\Omega)$, is harmonic in $\Omega$ and vanishes on $\Gamma$, then $u= 0$. These criteria play a central role in deriving regularity properties. Finally, we revisit the classical Area Integral Estimate. Using Grisvard's work and an explicit function given by Necas, we show that this inequality cannot hold in its stated form. Since this estimate has been widely used to argue that $H^{3/2}$-regularity is unattainable for data in the dual of $H^{1/2}_{00}(\Omega)$, our counterexample provides a decisive clarification.

☆ Hierarchical exact controllability for a parabolic equation with Hardy potential

Haiyang Lin, Bo You

The main objective of this paper is to study the hierarchical exact controllability for a parabolic equation with Hardy potential by Stackelberg-Nash strategy. In linear case, we employ Lax-Milgram theorem to prove the existence of an associated Nash equilibrium pair corresponding to a bi-objective optimal control problem for each leader, which is responsible for an exact controllability property. Then the observability inequality of a coupled parabolic system is established by using global Carleman inequalities, which results in the existence of a leader that drives the controlled system exactly to any prescribed trajectory. In semilinear case, we first prove the well-posedness of the coupled parabolic system to obtain the existence of Nash quasi-equilibrium pair and show that Nash quasi-equilibrium is equivalent to Nash equilibrium. Based on these results, we establish the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory by Leray-Schauder fixed point theorem.

☆ Global behavior of the energy to the hyperbolic equation of viscoelasticity with combined power-type nonlinearities

Haiyang Lin, Jinqi Yan, Bo You

The main objective of this manuscript is to investigate the global behavior of the solutions to the viscoelastic wave equation with a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as a supercritical source term which is a combined power-type nonlinearities. The global existence of the solutions is obtained provided that the energy sink dominates the energy source in an appropriate sense. In more general scenarios, we prove the global existence of the solutions if the initial history value $u_0$ is taken from a subset of a suitable potential well. Based on global existence results, the energy decay rate is derived which depends on the relaxation kernel as well as the growth rate of the damping term. In addition, we study blow-up of solutions when the source is stronger than dissipation.

☆ Strong convergence of fully discrete finite element schemes for the stochastic semilinear generalized Benjamin-Bona-Mahony equation driven by additive Wiener noise

Suprio Bhar, Mrinmay Biswas, Mangala Prasad

In this article, we have analyzed semi-discrete finite element approximation and full discretization of the Stochastic semilinear generalized Benjamin-Bona-Mahony equation in a bounded convex polygonal domain driven by additive Wiener noise. We use the finite element method for spatial discretization and the semi-implicit method for time discretization and derive a strong convergence rate with respect to both parameters (spatial and temporal). Numerical experiments have also been performed to support theoretical bounds.

comment: 20 pages, comments welcome!

☆ One-dimensional symmetry results for semilinear equations and inequalities on half-spaces

Nicolas Beuvin, Alberto Farina

We prove new one-dimensional symmetry results for non-negative solutions, possibly unbounded, to the semilinear equation $ -\Delta u= f(u)$ in the upper half-space $\mathbb{R}^{N}_{+}$. Some Liouville-type theorems are also proven in the case of differential inequalities in $\mathbb{R}^{N}_{+}$, even without imposing any boundary condition. Although subject to dimensional restrictions, our results apply to a broad family of functions $f$. In particular, they apply to all non-negative $f$ that behaves at least linearly at infinity.

☆ A Liouville theorem for the $2$-Hessian equation on the Heisenberg group

Wei Zhang, Qi Zhou

In this paper, we prove a Liouville theorem for the $2$-Hessian equation on the Heisenberg group $\mathbb{H}^n$. The result is obtained by choosing a suitable test function and using integration by parts to derive the necessary integral estimates.

comment: 17 pages

☆ On shape optimization with large magnetic fields in two dimensions

Vladimir Lotoreichik, Léo Morin

This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this conclusion. Our main result implies that if, for a bounded simply-connected planar domain, the n-th eigenvalue of the magnetic Dirichlet Laplacian with uniform magnetic field is smaller than the corresponding eigenvalue for a disk of the same area, then the Fraenkel asymmetry of that domain tends to zero in the strong magnetic field limit. Comparable results are also derived for the magnetic Dirichlet Laplacian on rectangles, as well as the magnetic Dirac operator with infinite mass boundary conditions on smooth domains. As part of our analysis, we additionally provide a new estimate for the torsion function on rectangles.

comment: 15 pages

☆ On anisotropic energy conservation criteria of incompressible fluids

Yanqing Wang, Wei Wei, Yulin Ye

In this paper, by means of divergence-free condition, we establish an anisotropic energy conservation class enabling one component of velocity in the largest space $L^{3} (0,T; B^{1/3}_{3,\infty})$ for the 3D inviscid incompressible fluids, which extends the celebrated result obtained by Cheskidov, Constantin, Friedlander and Shvydkoy in [15, Nonlinearity 21 (2008)]. For viscous flows, we generalize famous Lions's energy conservation criteria to allow the horizontal components and vertical part of velocity to have different integrability.

comment: 25 pages

☆ A weak type $(p,a)$ criterion for operators, and applications

Bernhard Haak, El-Maati Ouhabaz

Let $(X, d, \mu)$ be a space of homogeneous type and $\Omega$ an open subset of $X$. Given a bounded operator $T: L^p(\Omega) \to L^q(\Omega)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for $p_0$ and $a$ such that $\frac{1}{p_0} - \frac{1}{a} = \frac{1}{p}-\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0, a)$ for Riesz potentials $L^{-\frac{\alpha}{2}}$ or for Riesz transform type operators $\nabla \Delta^{-\frac{\alpha}{2}}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)$ when the heat kernel of $L$ satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space $H^1_L$ associated with $L$ into $L^a(X)$. By duality this gives boundedness from $L^{a'}(X)$ into $\text{BMO}_L$.

comment: 25 pages

☆ Error estimates in the non-relativistic limit for the two-dimensional cubic Klein-Gordon equation

Yong Lu, Fangzheng Huang

In this paper, we study the non-relativistic limit of the two-dimensional cubic nonlinear Klein-Gordon equation with a small parameter $0<\varepsilon \ll 1$ which is inversely proportional to the speed of light. We show the cubic nonlinear Klein-Gordon equation converges to the cubic nonlinear Schr\"{o}dinger equation with a convergence rate of order $O(\varepsilon^2)$. In particular, for the defocusing case with high regularity initial data, we show error estimates of the form $C(1+t)^N \varepsilon^2$ at time $t$ up to a long time of order $\varepsilon^{-\frac{2}{N+1}}$, while for initial data with limited regularity, we also show error estimates of the form $C(1+t)^M\varepsilon$ at time $t$ up to a long time of order $\varepsilon^{-\frac{1}{M+1}}$. Here $N$ and $M$ are constants depending on initial data. The idea of proof is to reformulate nonrelativistic limit problems to stability problems in geometric optics, then employ the techniques in geometric optics to construct approximate solutions up to an arbitrary order, and finally, together with the decay estimates of the cubic Schr\"{o}dinger equation, derive the error estimates.

comment: 29 pages

☆ Two-dimensional steady supersonic ramp flows of Bethe-Zel'dovich-Thompson fluids

Geng Lai

Two-dimensional steady supersonic ramp flows are important and well-studied flow patterns in aerodynamics. Vimercati, Kluwick and Guardone [J. Fluid Mech., 885 (2018) 445--468] constructed various self-similar composite wave solutions to the supersonic flow of Bethe-Zel'dovich-Thompson (BZT) fluids past compressible and rarefactive ramps. We study the stabilities of the self-similar fan-shock-fan and shock-fan-shock composite waves constructed by Vimercati et al. in that paper. %In order to study the stabilities of the composite waves, we solve some classes of shock free boundary problems. In contrast to ideal gases, the flow downstream (or upstream) of a shock of a BZT fluid may possibly be sonic in the sense of the flow velocity relative to the shock front. In order to study the stabilities of the composite waves, we establish some a priori estimates about the type of the shocks and solve some classes of sonic shock free boundary problems. We find that the sonic shocks are envelopes of one out of the two families of wave characteristics, and not characteristics. This results in a fact that the flow downstream (or upstream) a sonic shock is not $C^1$ smooth up to the shock boundary. We use a characteristic decomposition method and a hodograph transformation method to overcome the difficulty cased by the singularity on sonic shocks, and derive several groups of structural conditions to establish the existence of curved sonic shocks.

comment: 55 pages, 16 figures

☆ Complex dynamics and pattern formation in a diffusive epidemic model with an infection-dependent recovery rate

Wael El Khateeb, Chanaka Kottegoda, Chunhua Shan

A diffusive epidemic model with an infection-dependent recovery rate is formulated in this paper. Multiple constant steady states and spatially homogeneous periodic solutions are first proven by bifur cation analysis of the reaction kinetics. It is shown that the model exhibits diffusion-driven instability, where the infected population acts as an activator and the susceptible population functions as an in hibitor. The faster movement of the susceptible class will induce the spatial and spatiotemporal patterns, which are characterized by k-mode Turing instability and (k1,k2)-mode Turing-Hopf bifurcation. The transient dynamics from a purely temporal oscillatory regime to a spatial periodic pattern are discov ered. The model reveals key transmission dynamics, including asynchronous disease recurrence, spa tially patterned waves, and the formation of localized hotspots. The study suggests that spatially targeted strategies are necessary to contain disease waves that vary regionally and cyclically.

☆ Numerical Approximation and Bifurcation Results for an Elliptic Problem with Superlinear Subcritical Nonlinearity on the Boundary

Shalmali Bandyopadhyay, Thomas Lewis, Dustin Nichols

We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial \eta} = \lambda f(u)$ on $\partial\Omega$ as well as an extension to a corresponding system of equations. While existence, uniqueness, nonexistence, and multiplicity results for such problems are well-established, their numerical treatment presents computational challenges due to the absence of comparison principles and complex bifurcation phenomena. We present finite difference formulations for both single equations and coupled systems with cross-coupling boundary conditions, establishing admissibility results for the finite difference method. We derive principal eigenvalue analysis for the linearized problems to determine unique bifurcation points from trivial solutions. The eigenvalue analysis provides additional insight into the theoretical properties of the problem while also providing intuition for computing approximate solutions based on the proposed finite difference formulation. We combine our finite difference methods with continuation methods to trace complete bifurcation curves, validating established existence and uniqueness results and consistent with the results of the principle eigenvalue analysis.

☆ Logarithmic wave decay for short range wavespeed perturbations with radial regularity

Gayana Jayasinghe, Katrina Morgan, Jacob Shapiro, Mengxuan Yang

We establish logarithmic local energy decay for wave equations with a varying wavespeed in dimensions two and higher, where the wavespeed is assumed to be a short range perturbation of unity with mild radial regularity. The key ingredient is H\"older continuity of the weighted resolvent for real frequencies $\lambda$, modulo a logarithmic remainder in dimension two as $\lambda \to 0$. Our approach relies on a study of the resolvent in two distinct frequency regimes. In the low frequency regime, we derive an expansion for the resolvent using a Neumann series and properties of the free resolvent. For frequencies away from zero, we establish a uniform resolvent estimate by way of a Carleman estimate.

comment: 33 pages

☆ Complex dynamics and pattern formation in a diffusive epidemic model with an infection-dependent recovery rate

Wael El Khateeb, Chanaka Kottegoda, Chunhua Shan

A diffusive epidemic model with an infection-dependent recovery rate is formulated in this paper. Multiple constant steady states and spatially homogeneous periodic solutions are first proven by bifurcation analysis of the reaction kinetics. It is shown that the model exhibits diffusion-driven instability, where the infected population acts as an activator and the susceptible population functions as an in hibitor. The faster movement of the susceptible class will induce the spatial and spatiotemporal patterns, which are characterized by k-mode Turing instability and (k1,k2)-mode Turing-Hopf bifurcation. The transient dynamics from a purely temporal oscillatory regime to a spatial periodic pattern are discovered. The model reveals key transmission dynamics, including asynchronous disease recurrence, spatially patterned waves, and the formation of localized hotspots. The study suggests that spatially targeted strategies are necessary to contain disease waves that vary regionally and cyclically.

☆ Zeroes of Eigenfunctions of Schrödinger Operators after Schwartzman

Willie Wai-Yeung Wong

Consider a complete, connected, smooth, oriented Riemannian manifold $(M,g)$ with boundary, such that the first Betti number vanishes. Sol Schwartzman proved that for Schr\"odinger operators of the form $-\Delta_g + V$ where $\Im(V)$ is signed, if $f: M\to\mathbb{C}$ is a non-vanishing element of its kernel, then $f$ has constant phase. The proof relied on dynamical systems methods applied to the gradient flow of the phase of $f$. In this manuscript we provide a more direct PDE argument that proves strengthened versions of the same facts.

♻ ☆ From Navier-Stokes to BV solutions of the barotropic Euler equations

Geng Chen, Moon-Jin Kang, Alexis F. Vasseur

In the realm of mathematical fluid dynamics, a formidable challenge lies in establishing inviscid limits from the Navier-Stokes equations to the Euler equations, wherein physically admissible solutions can be discerned. The pursuit of solving this intricate problem, particularly concerning singular solutions, persists in both compressible and incompressible scenarios. This article focuses on small $BV$ solutions to the barotropic Euler equation in one spatial dimension. Our investigation demonstrates that these solutions are inviscid limits for solutions to the associated compressible Navier-Stokes equation. Moreover, we extend our findings by establishing the well-posedness of such solutions within the broader class of inviscid limits of Navier-Stokes equations with locally bounded energy initial values.

comment: We fixed typos and added appendix B

♻ ☆ The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field

Fei Wang, Lingda Xu, Zeren Zhang

We address a stability threshold problem of the Couette flow $(y,0,0)$ in a uniform magnetic fleld $\alpha(\sigma,0,1)$ with $\sigma\in\mathbb{Q}$ for the 3D MHD equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. Previously, the authors in \cite{L20,RZZ25} obtained the threshold $\gamma=1$ for $\sigma\in\mathbb{R}\backslash\mathbb{Q}$ satisfying a generic Diophantine condition, where they also proved $\gamma = 4/3$ for a general $\sigma\in\mathbb{R}$. In the present paper, we obtain the threshold $\gamma=1$ in $H^N(N>13/2)$, hence improving the above results when $\sigma$ is a rational number. The nonlinear inviscid damping for velocity $u^2_{\neq}$ is also established. Moreover, our result shows that the nonzero modes of magnetic field has an amplification of order $\nu^{-1/3}$ even on low regularity, which is very different from the case considered in \cite{L20,RZZ25}.

♻ ☆ Commutativity and non-commutativity of limits in the nonlinear bending theory for prestrained microheterogeneous plates

Klaus Boehnlein, Lucas Bouck, Stefan Neukamm, David Padilla-Garza, Kai Richter

In this paper we study the derivation of nonlinear bending models for prestrained elastic plates from three-dimensional non-linear elasticity via homogenization and dimension reduction. We compare effective models obtained by either simultaneously or consecutively passing to the $\Gamma$-limits as the thickness $h\ll1$ and the size of the material microstructure $\e\ll1$ vanish. In the regime $\e\ll h$ we show that the consecutive and simultaneous limit are equivalent, and also analyze the rate of convergence. In contrast, we observe that there are several different limit models in the case $h\ll \e$.

♻ ☆ Unique continuation on planar graphs

Ahmed Bou-Rabee, William Cooperman, Shirshendu Ganguly

We show that a discrete harmonic function which is bounded on a large portion of a periodic planar graph is constant. A key ingredient is a new unique continuation result for the weighted graph Laplacian. The proof relies on the structure of level sets of discrete harmonic functions, using arguments as in Bou-Rabee--Cooperman--Dario (2023) which exploit the fact that, on a planar graph, the sub- and super-level sets cannot cross over each other. In the special case of the square lattice this yields a new, geometric proof of the Liouville theorem of Buhovsky--Logunov--Malinnikova--Sodin (2017).

comment: 12 pages, 5 figures; minor improvement of exposition

♻ ☆ $\mathrm{C}^2$ estimates for general $p$-Hessian equations on closed Riemannian manifolds

Yuxiang Qiao

We study the $\mathrm{C}^2$ estimates for $p$-Hessian equations with general left-hand and right-hand terms on closed Riemannian manifolds of dimension $n$. To overcome the constraints of closed manifolds, we advance a new kind of "subsolution", called pseudo-solution, which generalizes "$\mathcal{C}$-subsolution" to some extent and is well-defined for fully general $p$-Hessian equations. Based on pseudo-solutions, we prove the $\mathrm{C}^1$ estimates for general $p$-Hessian equations, and the corresponding second-order estimates when $p\in\{2, n-1, n\}$, under sharp conditions -- we don't impose curvature restrictions, convexity conditions or "MTW condition" on our main results. Some other conclusions related to a priori estimates and different kinds of "subsolutions" are also given, including estimates for "semi-convex" solutions and when there exists a pseudo-solution.

comment: Any comments welcome!

♻ ☆ Half-space decay for linear kinetic equations

Émeric Bouin, Stéphane Mischler, Clément Mouhot

We prove that solutions to linear kinetic equations in a half-space with absorbing boundary conditions decay for large times like $t^{-\frac{1}{2}-\frac{d}{4}}$ in a weighted $\sfL^{2}$ space and like $t^{-1-\frac{d}{2}}$ in a weighted $\sfL^{\infty}$ space, i.e. faster than in the whole space and in agreement with the decay of solutions to the heat equation in the half-space with Dirichlet conditions. The class of linear kinetic equations considered includes the linear relaxation equation, the kinetic Fokker-Planck equation and the Kolmogorov equation associated with the time-integrated spherical Brownian motion.

♻ ☆ Sobolev homeomorphisms and composition operators on homogeneous Lie groups

Alexander Ukhlov

In this article, we study Sobolev homeomorphisms and composition operators on homogeneous Lie groups. We prove that a measurable homeomorphism $\varphi: \Omega \to\widetilde{\Omega}$ belongs to the Sobolev space $L^{1}_{q}(\Omega; \widetilde{\Omega})$, $1\leq q < \infty$, if and only if $\varphi$ generates a bounded composition operator on Sobolev spaces.

comment: 14 pages

♻ ☆ Boundary Control for Wildfire Mitigation

Mohamed Camil Belhadjoudja, Mohamed Maghenem, Emmanuel Witrant, Didier Georges

In this paper, we propose a feedback control strategy to protect vulnerable areas from wildfires. We consider a system of coupled partial differential equations (PDEs) that models heat propagation and fuel depletion in wildfires and study two cases. First, when the wind velocity is known, we design a Neumann-type boundary controller guaranteeing that the temperature of some protected region converges exponentially, in the $L^2$ norm, to the ambient temperature. Second, when the wind velocity is unknown, we design an adaptive Neumann-type boundary controller guaranteeing the asymptotic convergence, in the $L^2$ norm, of the temperature of the protected region to the ambient temperature. In both cases, the controller acts along the boundary of the protected region and relies solely on temperature measurements along that boundary. Our results are supported by numerical simulations.

comment: Submitted to the 2025 IEEE Conference on Decision and Control (CDC'25)

♻ ☆ Existence and stability of the Riemann solutions for a non-symmetric Keyfitz--Kranzer type model

Rahul Barthwal, Christian Rohde, Anupam Sen

In this article, we develop a new hyperbolic model governing the first-order dynamics of a thin film flow under the influence of gravity and solute transport. The obtained system turns out to be a non-symmetric Keyfitz-Kranzer type system. We find an entire class of convex entropies in the regions where the system remains strictly hyperbolic. By including delta shocks, we prove the existence of unique solutions of the Riemann problem. We analyze their stability with respect to the perturbation of the initial data and to the gravity and surface tension parameters. Moreover, we discuss the large time behaviour of the solutions of the perturbed Riemann problem and prove that the initial Riemann states govern it. Thus, we confirm the structural stability of the Riemann solutions under the perturbation of initial data. Finally, we validate our analytical results with well-established numerical schemes for this new system of conservation laws.

♻ ☆ The nonlinear Schrödinger equation with sprinkled nonlinearity

Benjamin Harrop-Griffiths, Rowan Killip, Monica Visan

We prove global well-posedness for the cubic nonlinear Schr\"odinger equation with nonlinearity concentrated on a homogeneous Poisson process.

comment: 27 pages

♻ ☆ Topology of closed asymptotic curves on negatively curved surfaces

Mohammad Ghomi, Matteo Raffaelli

Motivated by Nirenberg's problem on isometric rigidity of tight surfaces, we study closed asymptotic curves $\Gamma$ on negatively curved surfaces $M$ in Euclidean $3$-space. In particular, using C\u{a}lug\u{a}reanu's theorem, we obtain a formula for the linking number $Lk(\Gamma,n)$ of $\Gamma$ with the normal $n$ of $M$. It follows that when $Lk(\Gamma, n)=0$, $\Gamma$ cannot have any locally star-shaped planar projections with vanishing crossing number, which extends observations of Kovaleva, Panov and Arnold. These results hold also for curves with nonvanishing torsion and their binormal vector field. Furthermore we construct an example where $n$ is injective but $Lk(\Gamma, n)\neq 0$, and discuss various restrictions on $\Gamma$ when $n$ is injective.

comment: 11 pages, 3 figures; Minor revisions; Accepted for publication in J. Geom. Anal

♻ ☆ Asymptotic Stability of multi-solitons for $1$d Supercritical NLS

Gong Chen, Abdon Moutinho

Consider the one-dimensional $L^2$ supercritical nonlinear Schr\"odinger equation \begin{equation} i\partial_{t}\psi+\partial^{2}_{x}\psi+\vert \psi\vert^{2k}\psi=0 \text{, $k>2$}. \end{equation} It is well known that solitary waves for this equation are unstable. In the pioneering work of Krieger and Schlag \cite{KriegerSchlag}, the asymptotic stability of a solitary wave was established on a codimension-one center-stable manifold. In the present paper, using linear estimates developed for one-dimensional matrix charge transfer models in our previous work, \cite{dispanalysis1}, we prove asymptotic stability of multi-solitons on a finite-codimension manifold for $k>\frac{11}{4}$, provided that the soliton velocities are sufficiently separated.

comment: Version 2,, 75 pages. Keywords: Scattering, Asymptotic Stability on $H^{1}$ norm, 1d supercritical NLS, multi-solitons, non-integrable, H1 norm, Center-stable Manifold; Mistype errors corrected; Comments are welcome

♻ ☆ A note on Wang's conjecture for harmonic functions with nonlinear boundary condition

Xiaohan Cai

We obtain some Liouville type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary and partially verifies Wang's conjecture (J. Geom. Anal. 31 (2021)). For the specific manifold $\mathbb{B}^n$, we present a new proof of this conjecture, which has been resolved by Gu-Li (Math. Ann. 391(2025)). Our proof is based on a general principle of applying the P-function method to such Liouville type results. As a further application of this method, we obtain some classification results for nonnegative solutions of some semilinear elliptic equations with a nonlinear boundary condition.

comment: 24 pages. Minor revisions. Comments are welcome!

♻ ☆ Holder regularity for fully nonlinear nonlocal equations

Juan Pablo Cabeza

In this survey we prove H\"older regularity results for viscosity solutions of fully nonlinear nonlocal uniformly elliptic second order differential equations with local gradient terms. This extends the nonlocal counterpart of the work of G. Barles, E. Chasseigne and C. Imbert in JEMS, 2011, to fully nonlinear extremal nonlocal operators.

☆ Strictly singular operators on the Baernstein and Schreier spaces

Niels Jakob Laustsen, JamesSmith

Every composition of two strictly singular operators is compact on the Baernstein space $B_p$ for $1 < p < \infty$ and on the $p$-convexified Schreier space $S_{p}$ for $1 \leq p < \infty$. Furthermore, every subsymmetric basic sequence in $B_p$ (respectively, $S_p$) is equivalent to the unit vector basis for $\ell_p$ (respectively, $c_0$), and the Banach spaces $B_p$ and $S_p$ contain block basic sequences whose closed span is not complemented.

☆ Some remarks on decay in countable groups and amalgamated free products

Srivatsav Kunnawalkam Elayavalli, Gregory Patchell, Lizzy Teryoshin

In this note, we first study the notion of subexponential decay (SD) for countable groups with respect to a length function, which generalizes the well-known rapid decay (RD) property, first discovered by Haagerup in 1979. Several natural properties and examples are studied, especially including groups that have SD, but not RD. This consideration naturally has applications in $C^*$-algebras. We also consider in this setting a permanence theorem for decay in amalgamated free products (proved also recently by Chatterji--Gautero), and demonstrate that it is in a precise sense optimal.

comment: 28 pages. Comments welcome

☆ On the dichotomy of $p$-walk dimensions on metric measure spaces

Meng Yang

On a volume doubling metric measure space endowed with a family of $p$-energies such that the Poincar\'e inequality and the cutoff Sobolev inequality with $p$-walk dimension $\beta_p$ hold, for $p$ in an open interval $I\subseteq (1,+\infty)$, we prove the following dichotomy: either $\beta_p=p$ for all $p\in I$, or $\beta_p>p$ for all $p\in I$.

comment: 21 pages

☆ Lotz-Peck-Porta and Rosenthal's theorems for spaces $C_p(X)$

Jerzy Kąkol, Ondřej Kurka, Wiesław Śliwa

For a Tychonoff space $X$ by $C_p(X)$ we denote the space $C(X)$ of continuous real valued functions on $X$ endowed with the pointwise topology. We prove that an infinite compact space $X$ is scattered if and only if every closed infinite-dimensional subspace in $C_p(X)$ contains a copy of $c_0$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$. This provides a $C_p$-version of the theorem of Lotz, Peck and Porta for Banach spaces $C(X)$ and $c_0$. Applications will be provided. We prove also a $C_p$-version of Rosenthal's theorem by showing that for an infinite compact $X$ the space $C_p(X)$ contains a closed copy of $c_{0}(\Gamma)$ (with the pointwise topology) for some uncountable set $\Gamma$ if and only if $X$ admits an uncountable family of pairwise disjoint open subsets of $X$. Illustrating examples, additional supplementing $C_p$-theorems and comments are included.

☆ Linear Convergence of Gradient Descent for Quadratically Regularized Optimal Transport

Alberto González-Sanz, Marcel Nutz, Andrés Riveros Valdevenito

In optimal transport, quadratic regularization is an alternative to entropic regularization when sparse couplings or small regularization parameters are desired. Here quadratic regularization means that transport couplings are penalized by the squared $L^2$ norm, or equivalently the $\chi^2$ divergence. While a number of computational approaches have been shown to work in practice, quadratic regularization is analytically less tractable than entropic, and we are not aware of a previous theoretical convergence rate analysis. We focus on the gradient descent algorithm for the dual transport problem in continuous and semi-discrete settings. This problem is convex but not strongly convex; its solutions are the potential functions that approximate the Kantorovich potentials of unregularized optimal transport. The gradient descent steps are straightforward to implement, and stable for small regularization parameter -- in contrast to Sinkhorn's algorithm in the entropic setting. Our main result is that gradient descent converges linearly; that is, the $L^2$ distance between the iterates and the limiting potentials decreases exponentially fast. Our analysis centers on the linearization of the gradient descent operator at the optimum and uses functional-analytic arguments to bound its spectrum. These techniques seem to be novel in this area and are substantially different from the approaches familiar in entropic optimal transport.

☆ A weak type $(p,a)$ criterion for operators, and applications

Bernhard Haak, El-Maati Ouhabaz

Let $(X, d, \mu)$ be a space of homogeneous type and $\Omega$ an open subset of $X$. Given a bounded operator $T: L^p(\Omega) \to L^q(\Omega)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for $p_0$ and $a$ such that $\frac{1}{p_0} - \frac{1}{a} = \frac{1}{p}-\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0, a)$ for Riesz potentials $L^{-\frac{\alpha}{2}}$ or for Riesz transform type operators $\nabla \Delta^{-\frac{\alpha}{2}}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)$ when the heat kernel of $L$ satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space $H^1_L$ associated with $L$ into $L^a(X)$. By duality this gives boundedness from $L^{a'}(X)$ into $\text{BMO}_L$.

comment: 25 pages

☆ Tensorial representations of positive weakly (q,r)-dominated multilinear operators

Abdelaziz Belaada, Adel Bounabab, Athmane Ferradi, Khalil Saadi

We introduce and study the class of positive weakly (q,r)-dominated multilinear operators between Banach lattices. This notion extends classical domination and summability concepts to the positive multilinear setting and generates a new positive multi-ideal. A Pietsch domination theorem and a polynomial version are established. Finally, we provide a tensorial representation that yields an isometric identification with the dual of an appropriate completed tensor product.

comment: 22 pages

☆ Continuous fragmentation equations in weighted $L^1$ spaces

Lyndsay Kerr, Wilson Lamb, Matthias Langer

We investigate an integro-differential equation that models the evolution of fragmenting clusters. We assume cluster size to be a continuous variable and allow for situations in which mass is not necessarily conserved during each fragmentation event. We formulate the initial-value problem as an abstract Cauchy problem (ACP) in an appropriate weighted $L^1$ space, and apply perturbation results to prove that a unique, physically relevant classical solution of the ACP is given by a strongly continuous semigroup for a wide class of initial conditions. Moreover, we show that it is often possible to identify a weighted $L^1$ space in which this semigroup is analytic, leading to the existence of a unique, physically relevant classical solution for all initial conditions belonging to that space. For some specific fragmentation coefficients, we provide examples of weighted $L^1$ spaces where our results can be applied.

☆ Generalized Blaschke--Santaló-type inequalities, without symmetry restrictions

Thomas A. Courtade, Edric Wang

Nakamura and Tsuji (2024) recently investigated a many-function generalization of the functional Blaschke--Santal\'o inequality, which they refer to as a generalized Legendre duality relation. They showed that, among the class of all even test functions, centered Gaussian functions saturate this general family of functional inequalities. Leveraging a certain entropic duality, we give a short alternate proof of Nakamura and Tsuji's result, and, in the process, eliminate all symmetry assumptions. As an application, we establish a Talagrand-type inequality for the Wasserstein barycenter problem (without symmetry restrictions) originally conjectured by Kolesnikov and Werner (\textit{Adv.~Math.}, 2022). An analogous geometric Blaschke--Santal\'o-type inequality is established for many convex bodies, again without symmetry assumptions.

comment: 15 pages. Comments welcome

☆ Operator realizations about a matrix-centre

Ali Karoobi, Robert T. W. Martin, Maximilian Tornes

We develop a general theory of operator realizations, or ``linear representations" of analytic functions in several non-commuting variables about a matrix-centre. In particular we show that a non-commutative function has a matrix-centre realization about any matrix tuple, $Y$, in its domain, if and only if it is a uniformly analytic non-commutative function defined in a uniformly open neighbourhood of $Y$. This extends the finite-dimensional realization theory of non-commutative rational functions maximally -- to all uniformly analytic non-commutative functions.

☆ Banach spaces with arbitrary finite Baire order

Anna Pelczar-Barwacz, Zdeněk Silber, Tomasz Wawrzycki

We investigate intrinsic Baire classes of Banach spaces defined by Argyros, Godefroy and Rosenthal (2003). We introduce a construction, for any Banach space $X$ with a basis, of an $\ell_1$-saturated separable Banach space $Y$ such that for any $\alpha \leqslant \omega_1$ we have $Y^{**}_{1+\alpha} \cong Y \oplus X^{**}_\alpha$, where $X^{**}_\alpha$ denotes the $\alpha$-th intrinsic Baire class of $X$. We apply this construction to answer two open problems by Argyros, Godefroy and Rosenthal (2003), namely we build separable Banach spaces of any Baire order less or equal to $\omega$, and a non-universal separable Banach space of order $\omega_1$. Finally, we apply the construction to show an analogue of a result of Lindenstrauss (1971) by constructing, for any Banach space $X$ with a basis and any $n \in \mathbb{N}$, a Banach space $Y$ such that $Y^{**}_n \cong Y^{**}_{n-1} \oplus X$, showing that any such $X$ can appear as the space of functionals in a bidual Banach space $Y^{**}$ that are of $n$-th intrinsic Baire class but not of $(n-1)$-th intrinsic Baire class.

♻ ☆ Topological Center of the Double Dual of the Orlicz Figà-Talamanca Herz Algebra

Arvish Dabra, N. Shravan Kumar

Let $G$ a locally compact group and $(\Phi,\Psi)$ be a complementary pair of Young functions. Let $A_\Phi(G)$ be the Orlicz analogue of the classical Fig\`{a}-Talamanca Herz algebra $A_p(G).$ In this article, we establish a necessary and sufficient condition for the equality $\Lambda(A_\Phi(G)^{\ast\ast}) = A_\Phi(G)$ to hold, where $\Lambda(A_\Phi(G)^{\ast\ast})$ denotes the topological center of the double dual of $A_\Phi(G)$ when equipped with the first Arens product. Furthermore, we prove several results concerning the semi-simplicity of the Banach algebras $A_\Phi(G)^{\ast\ast}$ and $UCB_\Psi(\widehat{G})^\ast.$

comment: 17 pages

♻ ☆ On the polar of Schneider's difference body

Julián Haddad, Dylan Langharst, Galyna V. Livshyts, Eli Putterman

In 1970, Schneider introduced the $m$th-order extension of the difference body $DK$ of a convex body $K\subset\mathbb R^n$, the convex body $D^m(K)$ in $\mathbb R^{nm}$. He conjectured that its volume is minimized for ellipsoids when the volume of $K$ is fixed. In this work, we solve a dual version of this problem: we show that the volume of the polar body of $D^m(K)$ is maximized precisely by ellipsoids. For $m=1$ this recovers the symmetric case of the celebrated Blaschke-Santal\'o inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically \'a la Bourgain-Milman. We also consider a functional version.

comment: 31 pages, comments welcome. Updated presentation of some facts. Keywords: Schneider's conjecture, Blaschke-Santal\'o inequality, polarity

☆ Mean field control with absorption

Pierre Cardaliaguet, Joe Jackson, Panagiotis E. Souganidis

In this paper we study a mean field control problem in which particles are absorbed when they reach the boundary of a smooth domain. The value of the N-particle problem is described by a hierarchy of Hamilton-Jacobi equations which are coupled through their boundary conditions. The value function of the limiting problem; meanwhile, solves a Hamilton-Jacobi equation set on the space of sub-probability measures on the smooth domain, i.e. the space of non-negative measures with total mass at most one. Our main contributions are (i) to establish a comparison principle for this novel infinite-dimensional Hamilton-Jacobi equation and (ii) to prove that the value of the N-particle problem converges in a suitable sense towards the value of the limiting problem as N tends to infinity.

☆ Duality estimates for subdiffusion problems including time-fractional porous medium type equations

Arlúcio Viana, Patryk Wolejko, Rico Zacher

We prove duality estimates for time-fractional and more general subdiffusion problems. An important example is given by subdiffusive porous medium type equations. Our estimates can be used to prove uniqueness of weak solutions to such problems, and they allow to extend a key estimate from classical reaction-diffusion systems to the subdiffusive case. Besides concrete equations involving a Laplacian, we also consider abstract problems in a Hilbert space setting.

comment: 24 pages

☆ Variable Matrix-Weighted Besov Spaces

Dachun Yang, Wen Yuan, Zongze Zeng

In this article, applying matrix ${\mathcal A}_{p(\cdot),\infty}$ weights introduced in our previous work, we introduce the matrix-weighted variable Besov space via the matrix weight $W$ or the reducing operators ${\mathbb{A}}$ of order $p(\cdot)$ for $W$, Then we show that, defined either by the matrix weight $W$ or the reducing operators ${\mathbb{A}}$ of order $p(\cdot)$ for $W$, the matrix-weighted variable Besov spaces (respectively, the matrix-weighted variable Besov sequence spaces) are both equal. Next, we establish the $\varphi$-transform theorem for matrix-weighted variable Besov spaces and, using this, find that the definition of matrix-weighted variable Besov spaces is independent of the choice of $\varphi$. After that, for the further discussion of variable Besov spaces, we establish the theorem of almost diagonal operators and then, by using this, we establish the molecular characterization. Then, with applying the molecular characterization, we obtain the wavelet and atomic characterizations of matrix-weighted variable Besov spaces. Finally, as an application, we consider some classical operators. By using the wavelet characterization, we establish the trace operator and obtain the theorem of trace operators. Moreover, with applying the molecular characterization, we establish the theorem of Calder\'on--Zygmund operators on matrix-weighted variable Besov spaces.

☆ Calibrated Reifenberg With Holes

Susanna Bertolini, Alessandro Preti, Daniele Valtorta

In this article, we study a calibrated version of Reifenberg theorem "with holes". In particular we study sets that are suitably approximable at all points and scales by calibrated planes and show that, without any additional hypotheses on $\beta$-numbers, this implies measure upper bounds and rectifiability. This article follows the main techniques introduced in a previous article, but it allows for holes in the sets under consideration, and is more self-contained.

☆ Non-homogeneous Schrodinger systems with sign-changing and general nonlinearities: Infinitely many solutions

Guanwei Chen

In this paper, we study the non-homogeneous nonlinear Schr\"{o}dinger system $$\left\{ \begin{array}{ll} -\triangle u_j+V_j(x) u_j=g_j(x,u_1,\cdots,u_m)+h_j(x),& x\in \Omega,\\ \\ u_j:=u_j(x)=0,& x\in \partial\Omega,\\ \\ j=1,2,\cdots,m, \end{array}\right. $$ where $\Omega\subset\mathbb{R}^{N}$ ($N\ge2$) is a bounded smooth domain, $(g_1,\cdots,g_m)$ is the gradient of $G(x,U)\in C^1(\Omega\times\mathbb{R}^m,\mathbb{R})$, $G(x,U)$ may be sign-changing, and it is super-quadratic or asymptotically-quadratic as $|U|\to\infty$. We obtain infinitely many solutions by using variational methods and perturbation methods, and we provide several typical examples to illustrate the main results. The {\bf main novelties} are as follows. (1) The nonlinearity $G$ may be sign-changing. (2) The nonlinearity $G$ is not only general, but also super-quadratic or asymptotically-quadratic at infinity and zero. (3) The nonlinearity $G$ is power-type or non-power-type. (4) We not only construct some new conditions, but also apply some conditions used in homogeneous problems to the study of non-homogeneous systems for the first time. The {\bf main difficulties} come from the following three aspects. (1) The proof of boundedness for $(PS)$ sequence of approximate functionals. (2) The detailed analysis of the asymptotic behaviors of approximate functionals. (3) The estimate of the upper and lower bounds for the minimax value sequence $\{c_k\}$ of the even function.

comment: 24 pages, 0 figures

☆ Singularidades para as soluções das Equações de Navier-Stokes and Euler e o Problema do Milênio

Milton C. Lopes Filho, Helena J. Nussenzveig Lopes

The purpose of this note is to offer a birds-eye view on the history and the state-of-the-art in the research surrounding the Millenium Prize problem for the Navier-Stokes equations, the general problem of singularities in fluid dynamics and the corresponding problem for the Euler equations. This is the content of a plenary talk delivered at the 2024 Biannual meeting of the Brazilian Math Society by Helena Nussenzveig Lopes and it is written in portuguese.

comment: in Portuguese language. Notas de uma palestra plen\'{a}ria, ministrada na XI Bienal de Matem\'{a}tica, S\~{a}o Carlos, Julho 2024. Uma vers\~{a}o em ingl\^{e}s est\'{a} submetida aos Anais do congresso

☆ A gradient estimate for the linearized translator equation

Kyeongsu Choi, Robert Haslhofer, Or Hershkovits

In this paper, we develop some analytic foundations for the linearized translator equation in $\mathbb{R}^4$, i.e. in the first dimension where the Bernstein property fails. This equation governs how the (noncompact) singularity models of the mean curvature flow in $\mathbb{R}^4$ fit together in a common moduli space. Here, we prove a gradient estimate, which gives a sharp bound for $W_v$, namely for the derivative of the variation field $W$ in the tip region. This serves as a substitute for the fundamental quadratic concavity estimate from Angenent-Daskalopoulos-Sesum, which has been crucial for controlling $Y_v$, namely the derivative of the profile function $Y$ in the tip region. Moreover, together with interior estimates by virtue of the linearized translator equation our gradient estimate implies a bound for $W_\tau$ as well. Hence, our gradient estimate also serves as substitute Hamilton's Harnack inequality, which has played an important role for controlling $Y_\tau$ in the tip region.

comment: 19 pages

☆ Normalized solutions for fractional Choquard equation with critical growth on bounded domain

Divya Goel, Asmita Rai

In this work, we establish the multiplicity of positive solutions for the following critical fractional Choquard equation with a perturbation on the star-shaped bounded domain $$ \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda u +\alpha|u|^{p-2}u+ \left( \int\limits_{\Omega} \frac{|u(y)|^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy\right) |u|^{2^{*}_{\mu ,s}-2}u\; \text{in} \; \Omega,\\ u>0\; \text{in}\; \Omega,\; \\ u = 0\; \text{in} \; \mathbb{R}^{N}\backslash\Omega, \\ \int_{\Omega}|u|^2 dx=d, \end{array} \right. $$ where, $s\in(0,1), N>2s$, $\alpha\in \mathbb{R}$, $d>0$, $2

☆ Normalized solution to Kirchhoff-fractional system involving critical Choquard nonlinearity

Divya Goel, Shilpa Gupta, Asmita Rai

In this article, we explore the fractional Kirchhoff-Choquard system given by $$ \left\{ \begin{array}{lr} (a+b\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} u|^2\;dx)(-\Delta)^su=\lambda_1u+(I_{\mu}*|v|^{{2^*_{\mu,s}}})|u|^{{2^*_{\mu,s}}-2}u +\alpha p (I_{\mu}*|v|^{q})|u|^{p-2}u \;\text{in}\;\mathbb{R}^N,\\ (a+b\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} v|^2\;dx)(-\Delta)^sv=\lambda_2v+ (I_{\mu}*|u|^{{2^*_{\mu,s}}})|v|^{{2^*_{\mu,s}}-2}u +\alpha q(I_{\mu}*|u|^{p})|v|^{q-2}v \;\;\text{in}\;\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2=d_1^2,\;\;\int_{\mathbb{R}^N}|v|^2=d_2^2. \end{array} \right. $$ where $N> 2s$, $s \in (0,1)$, $\mu \in (0, N)$, $\alpha \in\mathbb{R}$. Here, $I_{\mu}:\mathbb{R}^N \to \mathbb{R}$ denotes the Riesz potential. We denote by $2_{\mu,*}:=\frac{2N-\mu}{N}$ and $\frac{2N-\mu}{N-2s}:={2^*_{\mu,s}}$, the lower and upper Hardy-Littlewood-Sobolev critical exponents, repectively, and assume that $2_{\mu,*} < p,q< {2^*_{\mu,s}}$. Our primary focus is on the existence of normalized solutions for the case $\alpha>0$ in two scenarios: the $L^2$ subcritical case characterized by $22_{\mu,*}

☆ Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networks

Liya Gaynutdinova, Martin Doškář, Ondřej Rokoš, Ivana Pultarová

Physics-informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs) relevant to multiscale modeling, but they often fail when applied to materials with discontinuous coefficients, such as media with piecewise constant properties. This paper introduces a dual formulation for the PINN framework to improve the reliability of the homogenization of periodic thermo-conductive composites, for both strong and variational (weak) formulations. The dual approach facilitates the derivation of guaranteed upper and lower error bounds, enabling more robust detection of PINN failure. We compare standard PINNs applied to smoothed material approximations with variational PINNs (VPINNs) using both spectral and neural network-based test functions. Our results indicate that while strong-form PINNs may outperform VPINNs in controlled settings, they are sensitive to material discontinuities and may fail without clear diagnostics. In contrast, VPINNs accommodate piecewise constant material parameters directly but require careful selection of test functions to avoid instability. Dual formulation serves as a reliable indicator of convergence quality, and its integration into PINN frameworks enhances their applicability to homogenization problems in micromechanics.

☆ A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators

Ben Andrews, Daniel Hauer, Jessica Slegers

In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the linear Schr\"odinger equation on a Riemannian manifold $M$ with nonnegative Ricci curvature, where the potential term $V$ is bounded from below. Our approach is based on a multi-point maximum principle argument. Standard proofs of this result (see, for instance, Li-Yau [Acta Math, 1986]) rely on first establishing a gradient estimate. This requires the solution to be at least $C^4$ on $M$. We instead prove the Harnack inequality directly, which has the advantage of avoiding higher-order derivatives of the solution in the proof, enabling us to assume it is only $C^2$ on $M$. In the particular case that $V$ is the quadratic potential $V(x)=|x|^2$ and $M$ is the Euclidean space $\mathbb{R}^d$, we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schr\"odinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator $\Delta - x\cdot \nabla$ with quadratic potential in $\mathbb{R}^d$.

☆ Fractional Sobolev logarithmic inequalities

Vivek Sahu

We establish new logarithmic Sobolev inequalities in the fractional Sobolev setting. Our approach relies on a interpolation inequality, which can be viewed as a fractional Caffarelli-Kohn-Nirenberg inequality without weights. We further relate the optimal constant in this interpolation inequality to a corresponding variational problem. These results extend classical logarithmic Sobolev inequalities to the nonlocal framework and provide new tools for analysis in fractional Sobolev spaces.

comment: 10 pages

☆ Semialgebric rank-one convex hulls: 2x2 triangular matrices and beyond

Chiara Meroni, Bogdan Raita

We prove that the rank-one convex hull of finitely many $2\times 2$ triangular matrices is a semialgebraic set, defined by linear and quadratic polynomials. We present explicit constructions for five-point configurations and offer evidence suggesting that a similar characterization does not hold in the more general setting of directional convexity.

comment: 18 pages, comments are welcome!

☆ Sharp multiscale control for high order nonlinear equations

Frédéric Robert

We analyze the behavior of families $(u_\alpha)_{\alpha>0}$ of solutions to the high-order critical equation $P_\alpha u_\alpha=\Delta_g^k u_\alpha +\hbox{lot}=|u_\alpha|^{2^\star-2}u_\alpha$ on a Riemannian manifold $M$, with a uniform bound on the Dirichlet energy. We prove a sharp pointwise control of the $u_\alpha$'s by a sum of bubbles uniformly with respect to $\alpha\to +\infty$, that is $|u_\alpha|\leq C\Vert u_\infty \Vert_\infty +C\sum_{i=1}^NB_{i,\alpha}$ where $u_\infty \in C^{2k}(M)$ and the $(B_{i,\alpha})_\alpha$, $i=1,...,N$ are explicit standard peaks.

comment: 38 pages

☆ Unveiling Biological Models Through Turing Patterns

Yuhan Li, Hongyu Liu, Catharine W. K. Lo

Turing patterns play a fundamental role in morphogenesis and population dynamics, encoding key information about the underlying biological mechanisms. Yet, traditional inverse problems have largely relied on non-biological data such as boundary measurements, neglecting the rich information embedded in the patterns themselves. Here we introduce a new research direction that directly leverages physical observables from nature--the amplitude of Turing patterns--to achieve complete parameter identification. We present a framework that uses the spatial amplitude profile of a single pattern to simultaneously recover all system parameters, including wavelength, diffusion constants, and the full nonlinear forms of chemotactic and kinetic coefficient functions. Demonstrated on models of chemotactic bacteria, this amplitude-based approach establishes a biologically grounded, mathematically rigorous paradigm for reverse-engineering pattern formation mechanisms across diverse biological systems.

comment: 22 pages keywords: inverse reaction-diffusion equations, Turing patterns, Turing instability, periodic solutions, sinusoidal form

☆ Nonlocal Harnack inequalities for nonlocal double phase equations I ; with positive bounded modulating coefficient with no Hölder condition

Yong-Cheol Kim

In this paper, by applying the De Giorgi-Nash-Moser theory we prove nonlocal Harnack inequalities for (locally nonnegative in $\Omega$) weak solutions to nolocal double phase equations \begin{equation*}\begin{cases}\cL u =0 & \text{ in $\Omega$,} \\ u=g & \text{ in $\BR^n\s\Omega$ } \end{cases}\end{equation*} where $\Omega\subset\BR^n$ ($n\ge 2$) is a bounded domain with Lipschitz boundary, $\cL$ is the nonlocal double phase operator $\cL$ given by \begin{equation*}\begin{split}\cL u(x)=&\pv\int_{\BR^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{ps}(x,y)\,dy \\ &+\pv\int_{\BR^n}\fa(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{qt}(x,y)\,dy, \end{split} \end{equation*} $0<\fa(x,y) = \fa(y,x) \le \|\fa\|_{L^\iy(\BR^n\times\BR^n)} < \iy$ and $ps\ge qt$ for $0

comment: 43 pages

☆ How smooth are restrictions of Besov functions?

Julien Brasseur

In a previous work, we showed that Besov spaces do not enjoy the restriction property unless $q\leq p$. Specifically, we proved that if $p

comment: Preliminary version: a longer version with additional results will follow. Comments are welcome

☆ Blow-up for a Nonlocal Diffusion Equation with Time Regularly Varying Nonlinearity and Forcing

Rihab Ben Belgacem, Mohamed Majdoub

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho \mathbf{w}(x), \qquad (x,t)\in \mathbb{R}^N\times (0,\infty), \] where $\mathsf{s}\in (0,1)$, $p>1$, $b\geq 0$, and $\varrho>-1$. The function $\mathsf{h}(t)$ is assumed to belong to the generalized class of regularly varying functions, while $\mathbf{w}$ is a prescribed spatial source. We first revisit the unforced case and establish sharp blow-up and global existence criteria in terms of the critical Fujita exponent, thereby extending earlier results to the wider class of time-dependent coefficients. For the forced problem, we derive nonexistence of global weak solutions under suitable growth conditions on $\mathsf{h}$ and integrability assumptions on $\mathbf{w}$. Furthermore, we provide sufficient smallness conditions on the initial data and the forcing term ensuring global-in-time mild solutions. Our analysis combines semigroup estimates for the mixed operator, test function methods, and asymptotic properties of regularly varying functions. To our knowledge, this is the first study addressing blow-up phenomena for nonlinear diffusion equations with such a class of time-dependent coefficients.

comment: 21 pages. Comments and suggestions are most welcome

☆ Existence and stability of the Riemann solutions for a non-symmetric Keyfitz--Kranzer type model

Rahul Barthwal, Christian Rohde, Anupam Sen

In this article, we develop a new hyperbolic model governing the first-order dynamics of a thin film flow under the influence of gravity and solute transport. The obtained system turns out to be a non-symmetric Keyfitz-Kranzer type system. We find an entire class of convex entropies in the regions where the system remains strictly hyperbolic. By including delta shocks, we prove the existence of unique solutions of the Riemann problem. We analyze their stability with respect to the perturbation of the initial data and to the gravity and surface tension parameters is analyzed. Moreover, we discuss the large time behaviour of the solutions of the perturbed Riemann problem and prove that the initial Riemann states govern it. Thus, we confirm the structural stability of the Riemann solutions under the perturbation of initial data. Finally, we validate our analytical results with well-established numerical schemes for this new system of conservation laws.

☆ On the exponential convergence to equilibrium for ultrafast diffusion equations

Yi C. Huang, Xinhang Tong

We propose a simple proof of the exponential convergence to equilibrium for ultrafast diffusion equations in $\mathbb{R}^n$. Our approach, based on the direct use of Poincar\'e inequality, gets rid of the optimal transport arguments used in \cite{fathi2025} which are valid for Gaussian-excluded one-dimensional weights. This simplification allows us to extend their results to Gaussian measures in higher dimensions.

comment: 5pages

☆ Gradient Flows of Interfacial Energies: Curvature Agents and Incompressibility

Keith Promislow, Truong Vu, Brian Wetton

We present a framework for the gradient flow of sharp-interface surface energies that couple to embedded curvature active agents. We use a penalty method to develop families of locally incompressible gradient flows that couple interface stretching or compression to local flux of interfacial mass. We establish the convergence of the penalty method to an incompressible flow both formally for a broad family of surface energies and rigorously for a more narrow class of surface energies. We present an analysis, including a $\Gamma$-limit, of an Allen-Cahn type model for a coupled surface agent curvature energy.

comment: 28 pages, 5 figures

☆ The bidirectional NLS approximation for the one-dimensional Euler-Poisson system

Huimin Liu, Yurui Lu, Xueke Pu

The nonlinear Schr\"{o}dinger (NLS) equation is known as a universal equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet in various dispersive systems. In this paper, we prove that under a certain multiple scale transformation, solutions to the Euler-Poisson system can be approximated by the sums of two counter-propagating waves solving the NLS equations. It extends the earlier results [Liu and Pu, Comm. Math. Phys., 371(2), (2019)357-398], which justify the unidirectional NLS approximation to the Euler-Poisson system for the ion-coustic wave. We demonstrate that the solutions could be convergent to two counter-propagating wave packets, where each wave packet involves independently as a solution of the NLS equation. We rigorously prove the validity of the NLS approximation for the one-dimensional Euler-Poisson system by obtaining uniform error estimates in Sobolev spaces. The NLS dynamics can be observed at a physically relevant timespan of order $\mathcal{O}(\epsilon^{-2})$. As far as we know, this result is the first construction and valid proof of the bidirectional NLS approximation.

comment: 56pages

☆ Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces

Jan Burczak, Antonio Hidalgo-Torné

For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p \in (0,1)$. Thus, in terms of the path space $C(\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains.

☆ Minimizing solutions of degenerate Allen-Cahn equations with three wells in $\mathbb{R}^2$

Lia Bronsard, Étienne Sandier, Peter Sternberg

We characterize all minimizers of the vector-valued Allen-Cahn equation in $\mathbb{R}^2$ under the assumption that the potential $W$ has three wells and that the associated degenerate metric does not satisfy the usual strict triangle inequality. These minimizers depend on one variable only in a suitable coordinate system. In particular, we show that no minimizing solutions to $ \Delta u=\nabla W(u)$ on $\mathbb{R}^2$ can approach the three distinct values of the potential wells.

☆ The non-relativistic limit of scattering states for the Vlasov equation with short-range interaction potentials

Younghun Hong, Stephen Pankavich

We study the relativistic and non-relativistic Vlasov equation driven by short-range interaction potentials and identify the large time dynamics of solutions. In particular, we construct global-in-time solutions launched from small initial data and prove that they scatter along the forward free flow to well-behaved limits as $t \to \infty$. Moreover, we prove the existence of wave operators for such a regime and, upon constructing the aforementioned time asymptotic limits, use the wave operator formulation to prove for the first time that the relativistic scattering states converge to their non-relativistic counterparts as $c \to \infty$.

comment: 33 pages

♻ ☆ The Hele-Shaw semi-flow

Thomas Alazard, Herbert Koch

We prove that the Cauchy problem is well-posed in a strong sense and in a general setting. Our main result is the construction of an abstract semi-flow for the Hele-Shaw problem within general fluid domains (enabling, for instance, changes in the topology of the fluid domain) and which satisfies several properties: We provide simple comparison arguments, establish a new stability estimate and derive several consequences, including monotonicity and continuity results for the solutions, along with many Lyapunov functionals. We establish an eventual analytic regularity result for any arbitrary initial data. We also study numerous qualitative properties, including global regularity for initial data in sub-critical Sobolev spaces, well-posedness in a strong sense for initial data with barely a modulus of continuity, as well as waiting-time phenomena for Lipschitz solutions, in any dimension. This revision contains some corrections.

comment: 79 pages

♻ ☆ On Limit Formulas for Besov Seminorms and Nonlocal Perimeters in the Dunkl Setting

Huaiqian Li, Bingyao Wu

We investigate the limiting behavior of Besov seminorms and nonlocal perimeters in Dunkl theory. The present work generalizes two fundamental results: the Maz'ya--Shaposhnikova formula for Gagliardo seminorms and the asymptotics of (relative) fractional $s$-perimeters. Our main contributions are twofold. First, we establish a dimension-free Maz'ya--Shaposhnikova formula via a novel, robust approach that avoids reliance on the density property of Besov spaces, offering broader applicability. Second, we prove limit formulas for nonlocal perimeters relative to bounded open sets $\Omega$, removing boundary regularity assumptions in the forward direction, while introducing a weakened regularity condition on $\partial\Omega$ (admitting fractal boundaries) for the converse, a significant improvement over existing requirements. To the best of our knowledge, the results in this second part are new even in the classic Laplacian setting.

comment: 39 pp, 1 figure

♻ ☆ Upwind filtering of scalar conservation laws

Giuseppe Maria Coclite, Kenneth Hvistendahl Karlsen, Nils Henrik Risebro

We study a class of multi-dimensional non-local conservation laws of the form $\partial_t u = \operatorname{div}^{\Phi} \mathbf{F}(u)$, where the standard local divergence $\operatorname{div}$ of the flux vector $\mathbf{F}(u)$ is replaced by an average upwind divergence operator $\operatorname{div}^{\Phi}$ acting on the flux along a continuum of directions given by a reference measure and a filter $\Phi$. The non-local operator $\operatorname{div}^{\Phi}$ applies to a general non-monotone flux $\mathbf{F}$, and is constructed by decomposing the flux into monotone components according to wave speeds determined by $\mathbf{F}'$. Each monotone component is then consistently subjected to a non-local derivative operator that utilizes an anisotropic kernel supported on the "correct" half of the real axis. We establish well-posedness, derive a priori and entropy estimates, and provide an explicit continuous dependence result on the kernel. This stability result is robust with respect to the "size" of the kernel, allowing us to specify $\Phi$ as a Dirac delta $\delta_0$ to recover entropy solutions of the local conservation law $\partial_t u = \operatorname{div} \mathbf{F}(u)$ (with an error estimate). Other choices of $\Phi$ (and the reference measure) recover known numerical methods for (local) conservation laws. This work distinguishes itself from many others in the field by developing a consistent non-local approach capable of handling non-monotone fluxes.

comment: Revised version, to appear in SIMA

♻ ☆ Generic regularity of equilibrium measures for the logarithmic potential with external fields

Giacomo Colombo, Alessio Figalli

It is a well-known conjecture in $\beta$-models and in their discrete counterpart that, generically, external potentials should be ``off-critical'' (or, equivalently, ``regular''). Exploiting the connection between minimizing measures and thin obstacle problems, we give a positive answer to this conjecture.

comment: 24 pages, comments are welcome

♻ ☆ Rates of convergence in long time asymptotics of an alignment model with symmetry breaking

Alexandre Surin

We consider a nonlinear Fokker-Planck equation derived from a Cucker-Smale model for flocking with noise. There is a known phase transition depending on the noise between a regime with a unique stationary solution which is isotropic (symmetry) and a regime with a continuum of polarized stationary solutions (symmetry breaking). If the value of the noise is larger than the threshold value, the solution of the evolution equation converges to the unique radial stationary solution. This solution is linearly unstable in the symmetry-breaking range, while polarized stationary solutions attract all solutions with sufficiently low entropy. We prove that the convergence measured in a weighted $L^2$ norm occurs with an exponential rate and that the average speed also converges with exponential rate to a unique limit which determines a single polarized stationary solution.

♻ ☆ Envelope representation of Hamilton-Jacobi equations from spin glasses

Hong-Bin Chen

Recently, [arXiv:2311.08980] demonstrated that, if it exists, the limit free energy of possibly non-convex spin glass models must be determined by a characteristic of the associated infinite-dimensional non-convex Hamilton-Jacobi equation. In this work, we investigate a similar theme purely from the perspective of PDEs. Specifically, we study the unique viscosity solution of the aforementioned equation and derive an envelope-type representation formula for the solution, in the form proposed by Evans in [doi:10.1007/s00526-013-0635-3]. The value of the solution is expressed as an average of the values along characteristic lines, weighted by a non-explicit probability measure. The technical challenges arise not only from the infinite dimensionality but also from the fact that the equation is defined on a closed convex cone with an empty interior, rather than on the entire space. In the introduction, we provide a description of the motivation from spin glass theory and present the corresponding results for comparison with the PDE results.

comment: 36 pages; journal version

♻ ☆ Nonlinear stability of 2-D Couette flow for the compressible Navier-Stokes equations at high Reynolds number

Minling Li, Chao Wang, Zhifei Zhang

In this paper, we investigate the nonlinear stability of the Couette flow for the two-dimensional compressible Navier--Stokes equations at high Reynolds numbers ($Re$) regime. It was proved that if the initial data $(\rho_{in},u_{in})$ satisfies $\|(\rho_{in},u_{in})-(1, y, 0)\|_{H^4(\mathbb{T}\times\mathbb{R})}\leq \epsilon Re^{-1}$ for some small $\epsilon$ independent of $Re$, then the corresponding solution exists globally and remains close to the Couette flow for all time. Formal asymptotics indicate that this stability threshold is sharp within the class of Sobolev perturbations. The proof relies on the Fourier-multiplier method and exploits three essential ingredients: (i) the introduction of ``good unknowns" that decouple the perturbation system; (ii) the construction of a carefully designed Fourier multiplier that simultaneously captures the enhanced dissipation and inviscid-damping effects while taming the lift-up mechanism; and (iii) the design of distinct energy functionals for the incompressible and compressible modes.

♻ ☆ Critical double phase problems involving sandwich-type nonlinearities

Csaba Farkas, Alessio Fiscella, Ky Ho, Patrick Winkert

In this paper we study problems with critical and sandwich-type growth represented by \begin{align*} -\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= \lambda w(x)|u|^{s-2}u+\theta B\left(x,u\right) \quad \text{in } \Omega,\quad u= 0 \quad\text{on } \partial \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partial\Omega$, $1

☆ Some remarks on $M_d$-multipliers and approximation properties

Ignacio Vergara

We prove an extension property for $M_d$-multipliers from a subgroup to the ambient group, showing that $M_{d+1}(G)$ is strictly contained in $M_d(G)$ whenever $G$ contains a free subgroup. Another consequence of this result is the stability of the $M_d$-approximation property under group extensions. We also show that Baumslag-Solitar groups are $M_d$-weakly amenable with $\boldsymbol\Lambda(\operatorname{BS}(m,n),d)=1$ for all $d\geq 2$. Finally, we show that, for simple Lie groups with finite centre, $M_d$-weak amenability is equivalent to weak amenability, and we provide some estimates on the constants $\boldsymbol\Lambda(G,d)$.

comment: 23 pages, comments welcome

☆ Variable Matrix-Weighted Besov Spaces

Dachun Yang, Wen Yuan, Zongze Zeng

In this article, applying matrix ${\mathcal A}_{p(\cdot),\infty}$ weights introduced in our previous work, we introduce the matrix-weighted variable Besov space via the matrix weight $W$ or the reducing operators ${\mathbb{A}}$ of order $p(\cdot)$ for $W$, Then we show that, defined either by the matrix weight $W$ or the reducing operators ${\mathbb{A}}$ of order $p(\cdot)$ for $W$, the matrix-weighted variable Besov spaces (respectively, the matrix-weighted variable Besov sequence spaces) are both equal. Next, we establish the $\varphi$-transform theorem for matrix-weighted variable Besov spaces and, using this, find that the definition of matrix-weighted variable Besov spaces is independent of the choice of $\varphi$. After that, for the further discussion of variable Besov spaces, we establish the theorem of almost diagonal operators and then, by using this, we establish the molecular characterization. Then, with applying the molecular characterization, we obtain the wavelet and atomic characterizations of matrix-weighted variable Besov spaces. Finally, as an application, we consider some classical operators. By using the wavelet characterization, we establish the trace operator and obtain the theorem of trace operators. Moreover, with applying the molecular characterization, we establish the theorem of Calder\'on--Zygmund operators on matrix-weighted variable Besov spaces.

☆ Loop space blow-up and scale calculus

Urs Frauenfelder, Joa Weber

In this note we show that the Barutello-Ortega-Verzini regularization map is scale smooth.

comment: 5 pages

☆ New examples of Fourier multipliers on $H^1\p{\mathbb{D}^2}$ revisited

Maciej Rzeszut

We show yet another family of examples of idempotent Fourier multipliers on $H^1\p{\mathbb{D}^2}$. The proof differs from our previous result and gets rid of arithmetical assumptions.

☆ Disjointification inequalities for Hoeffding subspaces of $H^1$ on an infinite polydisc

Maciej Rzeszut

We prove that the $L^1$ norm on the linear span of functions on $\T^\N$ dependent on $m$ variables and analytic and mean zero in each of them can be expressed as an interpolation sum of $H^1\left(\mathbb{T}^S,\ell^1\left(\mathbb{N}^S,H^2\left(\mathbb{T}^{[1,m]\setminus S},\ell^2\left(\mathbb{N}^{[1,m]\setminus S}\right)\right)\right)\right)$ norms over $S\subseteq [1,m]$ and derive some interpolation consequences.

☆ Fefferman multiplier theorem for Hardy martingales

Maciej Rzeszut

A well-known theorem due to Fefferman provides a characterization of Fourier multipliers from $H^1(\mathbb{T})$ to $\ell^1$, i.e. sequences $\left(\lambda_n\right)_{n=0}^\infty$ such that \[\sum_{n=0}^\infty \left|\lambda_n \widehat{f}(n)\right|\lesssim \|f\|_{L^1(\mathbb{T})},\] where $f(x)=\sum_{n=0}^\infty \widehat{f}(n)e^{inx}$. We extend it to the space $H^1\left(\mathbb{T}^\mathbb{N}\right)$ of Hardy martingales, i.e. the subspace of $L^1$ on the countable product $\mathbb{T}^\mathbb{N}$ consisting of all $f$ such that the differences $\Delta_nf=f_{n}-f_{n-1}$ of the martingale wrt the standard filtration generated by $f$ satisfy \[\left(t\mapsto \Delta_n f\left(x_1,\ldots,x_{n-1},t\right)\right)\in H^1(\mathbb{T}). \] The key ingredient is a theorem due to P. F. X. M\"uller stating that the classical Davis-Garsia decomposition \[\mathbb{E} \left(\sum_{n=0}^\infty \left|\Delta_n f\right|^2\right)^\frac{1}{2}\simeq \inf_{f=g+h} \mathbb{E}\sum_{n=0}^\infty \left|\Delta_n g\right|+ \mathbb{E}\left(\sum_{n=0}^\infty \mathbb{E}\left(\left|\Delta_n f\right|^2\mid \mathcal{F}_{n-1}\right)\right)^\frac{1}{2}\] may be done within the space of Hardy martingales.

☆ Duality of mixed norm spaces induced by radial one-sided doubling weight

Álvaro Miguel Moreno, José Ángel Peláez

For $00$ such that $$\int_r^1\omega(s)ds \leq C \int_{\frac{1+r}{2}}^1\omega(s)\,ds \,\, \text{for every}\,\, 0\leq r <1.$$ We describe the dual space of $A^{p,q}_\omega$ for every $0

☆ Weaving Information Packets

Ole Christensen, Hong Oh Kim, Rae Young Kim

The concept of weaving of frames for Hilbert spaces was introduced by Bemrose et al. in 2016. Two frames $\{f_k\}_{k\in I}, \{g_k\}_{k\in I}$ are woven if the ``mixed system" $\{f_k\}_{k\in \sigma} \cup \{g_k\}_{k\in I\setminus \sigma}$ is a frame for each index set $\sigma \subset I;$ that is, processing a signal using two woven frames yields a certain stability against loss of information. The concept easily extends to $N$ frames, for any integer $N>2.$ Unfortunately it is nontrivial to construct useful woven frames, and the literature is sparse concerning explicit constructions. In this paper we introduce so-called information packets, which contain as well frames as fusion frames as special case. The concept of woven frames immediately generalizes to information packets, and we demonstrate how to construct practically relevant woven information packets based on particular wavelet systems in $\ltr.$ Interestingly, we show that certain wavelet systems can be split into $N$ woven information packets, for any integer $N\ge 2.$ We finally consider corresponding questions for Gabor system in $\ltr,$ and prove that for any fixed $N\in \mn$ we can find a Gabor frame that can be split into $N$ woven information packets; however, in contrast to the wavelet case, the density conditions for Gabor system excludes the possibility of finding a single Gabor frame that works simultaneously for all $N\in \mn.$

☆ Strongly continuous fields of operators over varying Hilbert spaces

Ali BenAmor, Batu Güneysu, Thomas Kalmes, Peter Stollmann

After introducing a natural notion of continuous fields of locally convex spaces, we establish a new theory of strongly continuous families of possibly unbounded self-adjoint operators over varying Hilbert spaces. This setting allows to treat operator families defined on bundles of Hilbert spaces that are not locally trivial (such as e.g.~the tangent bundle of Wasserstein space), without referring to identification operators at all.

comment: 40 pages, comments are welcome

☆ How smooth are restrictions of Besov functions?

Julien Brasseur

In a previous work, we showed that Besov spaces do not enjoy the restriction property unless $q\leq p$. Specifically, we proved that if $p

comment: Preliminary version: a longer version with additional results will follow. Comments are welcome

☆ On the spectrum of the symmetric tensor products of certain Hilbert-space operators

Yuchi Yang, Yuanhang Zhang

This paper primarily investigates spectral properties of symmetric tensor products of Hilbert-space operators. For a unilateral weighted shift operator $S_w$, we present an algorithm to compute the point spectrum of its symmetric and antisymmetric tensor products with the adjoint $S_w^*$. Additionally, we analyze the symmetric tensor product of an injective unilateral weighted shift $S_\alpha$ and a diagonal operator $M$ on $l^2$, demonstrating that its point spectrum must be contained in $\{0\}$.

☆ Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces

Jan Burczak, Antonio Hidalgo-Torné

For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the $2$d Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p \in (0,1)$. Thus, in terms of the path space $C(\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains.

♻ ☆ SBV functions in Carnot-Carathéodory spaces

Marco Di Marco, Sebastiano Don, Davide Vittone

We introduce the space SBV$_X$ of special functions with bounded $X$-variation in Carnot-Carath\'eodory spaces and study its main properties. Our main outcome is an approximation result, with respect to the BV$_X$ topology, for SBV$_X$ functions.

comment: 26 pages

♻ ☆ On Limit Formulas for Besov Seminorms and Nonlocal Perimeters in the Dunkl Setting

Huaiqian Li, Bingyao Wu

We investigate the limiting behavior of Besov seminorms and nonlocal perimeters in Dunkl theory. The present work generalizes two fundamental results: the Maz'ya--Shaposhnikova formula for Gagliardo seminorms and the asymptotics of (relative) fractional $s$-perimeters. Our main contributions are twofold. First, we establish a dimension-free Maz'ya--Shaposhnikova formula via a novel, robust approach that avoids reliance on the density property of Besov spaces, offering broader applicability. Second, we prove limit formulas for nonlocal perimeters relative to bounded open sets $\Omega$, removing boundary regularity assumptions in the forward direction, while introducing a weakened regularity condition on $\partial\Omega$ (admitting fractal boundaries) for the converse, a significant improvement over existing requirements. To the best of our knowledge, the results in this second part are new even in the classic Laplacian setting.

comment: 39 pp, 1 figure

♻ ☆ Non-reflexivity of the Banach space $ΛBV^{(p)}$

Batoul S. Mortazavi-Samarin, Mehdi Rostami

In this paper, we show that the Waterman-Shiba space is non-reflexive. In fact, Prus-Wi\'sniowski and Ruckle, in \cite{1}, generalized the well-known fact that states the space of bounded variation functions is non-reflexive. Here, an improvement of that result is provided.

comment: 8 papges

♻ ☆ Pełczyński's property (V$^*$) in Lipschitz-free spaces

Ramón J. Aliaga, Eva Pernecká, Alicia Quero

We prove that Pelczy\'nski's property (V$^*$) is locally determined for Lipschitz-free spaces, and obtain several sufficient conditions for it to hold. We deduce that $\mathcal{F}(M)$ has property (V$^*$) when the complete metric space $M$ is locally compact and purely 1-unrectifiable, a Hilbert space, or belongs to a class of Carnot-Carath\'eodory spaces satisfying a bi-H\"older condition, including Carnot groups.

♻ ☆ Unconditional Schauder frames of exponentials and of uniformly bounded functions in $L^p$ spaces

Nir Lev, Anton Tselishchev

It is known that there is no unconditional basis of exponentials in the space $L^p(\Omega)$, $p \ne 2$, for any set $\Omega \subset \mathbb{R}^d$ of finite measure. This is a consequence of a more general result due to Gaposhkin, who proved that the space $L^p(\Omega)$ does not admit a seminormalized unconditional basis consisting of uniformly bounded functions. We show that the latter result fails if the word "basis" is replaced with "Schauder frame". On the other hand we prove that if $\Omega$ has nonempty interior then there are no unconditional Schauder frames of exponentials in the space $L^p(\Omega)$, $p \ne 2$.

♻ ☆ Characterizing the Kirkwood-Dirac positivity on second countable LCA groups

Matéo Spriet

We define the Kirkwood-Dirac quasiprobability representation of quantum mechanics associated with the Fourier transform over second countable locally compact abelian groups. We discuss its link with the Kohn-Nirenberg quantization of the phase space $G\times \widehat{G}$. We use it to argue that in this abstract setting the Wigner-Weyl quantization, when it exists, can still be interpreted as a symmetric ordering. Then, we identify all generalized (non-normalizable) pure states having a positive Kirkwood-Dirac distribution. They are, up to the natural action of the Weyl-Heisenberg group, Haar measures on closed subgroups. This generalizes a result known for finite abelian groups. We then show that the classical fragment of quantum mechanics associated with the Kirkwood-Dirac distribution is non-trivial if and only if the group has a compact connected component. Finally, we provide for connected compact abelian groups a complete geometric description of this classical fragment.

comment: 30 pages

♻ ☆ Construction of linearly independent and orthogonal functions in Hilbert function spaces via Wronski determinants

Athanasios Christou Micheas

Based on the Wronski determinant, we propose the construction of linearly independent orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of the functions under consideration, that satisfies minimal properties. Two applications are considered, including solutions to ordinary differential equations and the construction of basis functions. We also present a conjecture that connects the latter two concepts, which leads to what we call the Wronski basis.

comment: 17 pages; generalization of Gram-Schmidt orthonormalization; construction of basis functions; differential equations and Wronski basis; conjecture on bases in Hilbert spaces