Analysis of PDEs
☆ Regularity theorems for random elliptic operators on domains
Regularity theorems à la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general smooth domains the existing theory has until recently remained limited to Lipschitz estimates. We establish $C^{1,α}$ regularity results for random elliptic operators on bounded sufficiently smooth domains, as well as for scalar problems on convex polytopes. We, furthermore, prove a number of auxiliary results typically employed in the derivation of fluctuation bounds, such as a weighted Meyers estimate.
comment: 33 pages, The results in this article have been split off from the first version of arXiv:2403.12911. It is, in particular, a companion of arXiv:2403.12911v2
☆ A Posteriori Error Analysis of Runge-Kutta Discontinuous Galerkin Schemes with SIAC Post-Processing for Nonlinear Convection-Diffusion Systems
We develop reliable a posteriori error estimators for fully discrete Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems endowed with a convex entropy in multiple spatial dimensions on the flat torus T^d, with a focus on the convection-dominated regime. In order to use the relative entropy method, we reconstruct the numerical solution via tensor-product Smoothness-Increasing Accuracy-Conserving (SIAC) filtering which has superconvergence properties. We then derive reliable a posteriori error estimators for the difference between the entropy weak solution and the reconstruction, with constants that are uniform in the vanishing viscosity limit. Our numerical experiments show that the a posteriori error bounds converge with the same order as the error of the reconstructed numerical solution.
comment: 21 pages, 1 figure, 10 tables
☆ Blow-up analysis and extremal functions for nonlocal interaction functionals in dimension $N$
In this paper we study Moser-Trudinger type inequalities for some nonlocal energy functionals in presence of a logarithmic convolution potential, when the domain is a ball of $\mathbb{R}^N$ with $N \geq 2$. In particular, we perform a blow-up analysis to prove existence of extremal functions in the borderline case of critical growth. Using this, we extend the results in \cite{CiWeYu} to higher dimension and sharpen \cite{CC}.
☆ Frequency-dependent capacitance matrix formulation for Fabry-Pérot resonances. Part I: One-dimensional finite systems
We study scattering resonances of finite one-dimensional systems of high-contrast resonators beyond the subwavelength regime. Introducing a novel tridiagonal frequency-dependent capacitance matrix, we derive quantitative asymptotic expansions of the hybridized Fabry-Pérot resonant frequencies in terms of the material contrast parameter. The leading-order shifts are governed by the eigenvalues of this matrix, while the corresponding eigenmodes are approximated, to leading order, by trigonometric functions on selected spacings between resonators. Our results extend the use of discrete approximations as a powerful tool for characterizing the resonant properties of a system of high-contrast resonators at arbitrarily high frequencies.
comment: 42 pages, 12 figures
☆ Stochastically-constrained Koiter shell models
We derive a stochastically-constrained Koiter shell model in line with the SALT (Stochastic Advection by Lie Transport) approach introduced by Holm [Proc. A. 471 (2015)]. First, we deduce the stochastic partial differential equations for the generalised nonlinearly- and linearly-elastic Koiter shell models with their abstract functional derivatives for their corresponding membrane and flexural energies. We then present a prototype for the stochastically-constrained (simplified) linearised Koiter shell models that encodes all necessary information on stiffness due to shell curvature, bending stress, membrane stress, interior and surface forces, and more generally, stochastic buckling.
☆ Non-Ljusternik--Schnirelman eigenvalues of the pure $p$-Laplacian exist
An old and well-known open problem in the critical point theory asks whether, for some $p \neq 2$ and some bounded domain $Ω$, there exists a critical value of the $p$-Dirichlet energy $\|\nabla u\|_p^p$ over an $L^p(Ω)$-sphere in $W_0^{1,p}(Ω)$ lying outside of a Ljusternik--Schnirelman type sequence of critical values, the latter will be called LS eigenvalues of the $p$-Laplacian. In this work, we provide a positive answer by showing the existence of a non-LS eigenvalue when $p>2$ is sufficiently close to $2$ and $Ω$ is just a planar rectangle close to the square. The arguments pursue the observation that a simple eigenvalue of the Laplacian can be a meeting point for several branches of eigenvalues of the $p$-Laplacian as $p$ varies. Since LS eigenvalues are continuous with respect to $p$ and exhaust the whole spectrum when $p=2$, we deduce that at least one of the branches must contain non-LS eigenvalues.
comment: 13 pages, 1 figure
☆ Oscillations in a scalar differential equation coupled to a diffusive field
We study the emergence of periodic oscillations through a Hopf bifurcation in a scalar diffusion equation on the half line coupled to a dynamic boundary condition. Our results quantify the effect of delay through the buffering in the diffusive field on boundary kinetics, drawing a parallel to the emergence of oscillations in delay equations. Technically, the Hopf bifurcation occurs in the presence of essential spectrum induced by the diffusive field, preventing a simple approach via center-manifold reduction. The results are motivated by observations in biological systems where dynamic boundary conditions arise when modeling surface dynamics coupled to bulk diffusion.
comment: 23 pages, 5 figures
☆ Topological entropy is generically infinite for non-Lipschitz velocity fields
We prove that for any Osgood non-Lipschitz modulus of continuity $ω$, flow maps associated with time-periodic $ω$-continuous velocity fields generically (in the sense of Baire) have infinite topological entropy.
comment: 9 pages, 1 figure
☆ The Semiclassical Einstein-Klein-Gordon System: Asymptotic Analysis of Minkowski Spacetime
We establish the linear instability of the semiclassical Einstein-Klein-Gordon system linearised about the Minkowski vacuum spacetime. The proof relies on formulating a forcing problem for both metric and state perturbations within the space of past-compact sections. This geometric framework admits a unique tensor decomposition which, in conjunction with the quantum Møller operator, enables the decoupling of the linearised system into two distinct Cauchy problems. Consequently, the metric perturbations are shown to be governed by a higher-order, nonlocal hyperbolic partial differential equation. By relegating the nonlocal contributions to subleading order, we establish the well-posedness of this forcing problem. Furthermore, we provide a rigorous asymptotic analysis for physically admissible choices of the renormalisation constants. We prove that the system exhibits a late-time linear instability: the metric perturbations grow exponentially, bounded strictly by a universal scale H, thereby indicating a quantum backreaction-driven transition toward a de Sitter cosmological spacetime. Provided the parameters governing the system are restricted to a physically relevant regime, this universal scale is compatible with the measured expansion of our universe.
comment: 71 pages, 1 figure
☆ Rigorous $C_1$ integration of dissipative PDEs
We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.
☆ On the Effectless Cut Method for Laplacian Eigenvalues in any dimensions
In this paper, we study the optimization of the first Laplacian eigenvalue on axisymmetric doubly connected domains under positive Robin boundary conditions. Under additional geometric constraints, we prove that spherical shells maximize this eigenvalue. Our approach combines known isoperimetric inequalities for mixed Laplacian eigenvalues with a higher-dimensional extension of the effectless cut technique introduced by Hersch to study multiply connected membranes of given area fixed along their boundaries.
☆ Maximal regularity for a compressible fluid-structure interaction system with Navier-slip boundary conditions
We investigate a fluid-structure interaction system in which the dynamics of the fluid is described by the compressible Navier-Stokes equations, while the elastic structure is modeled by a damped plate equation. The fluid evolves in a three-dimensional bounded domain, with the structure occupies a part of its boundary. Instead of standard no-slip boundary conditions, we consider the Navier-slip boundary conditions at the fluid-structure interface as well as at the fixed boundary. We establish the local-in-time existence and uniqueness of strong solutions within $L^{p}-L^{q}$ framework. The existence result is obtained for small time by decoupling the linearized system and employing a cascade strategy combined with the Tikhonov fixed point theorem, whereas the uniqueness is shown by deriving weak regularity properties for the associated linear coupled operator in a Hilbert space setting. It is the first result addressing strong solutions for a compressible fluid interacting with a damped plate under Navier-slip boundary conditions.
☆ On the equivalence of generalized solution concepts for systems of hyperbolic conservations laws in fluid dynamics
We investigate the relation between several generalized solution concepts for nonlinear PDE systems from fluid dynamics. More precisely, we study measure-valued solutions, dissipative weak solutions, and energy-variational solutions. For the incompressible Euler equations, we prove the equivalence of all three concepts, provided that the energy inequality is formulated in the appropriate way. For several important examples of conservation laws arising in fluid dynamics, we establish the equivalence between energy-variational and suitably refined dissipative weak solutions, where the defect measures are controlled sharply by the energy defect. These examples comprise the compressible isentropic Euler system, the Euler--Korteweg system, and the Euler--Poisson system.
☆ Sharp local sparsity of regularized optimal transport
In recent years, the use of entropy-regularized optimal transport with $L^p$-type entropies has become increasingly popular. In this setting, the solutions are sparse, in the sense that the support of the regularized optimal coupling, $\mathrm{supp}(π_\varepsilon)$, shrinks to the support of the original optimal transport problem as $\varepsilon \to 0$.
The main open question concerns the rate of this convergence. In this paper, we obtain sharp local results away from the boundary. We prove that the supports $\mathrm{supp}(π_\varepsilon(\cdot \mid x))$ of the conditional measures, $π_\varepsilon(\cdot \mid x)$, behave like balls of radius $\varepsilon^\frac 1 {d(p-1)+2}$. This allows us to show that the regularized potentials are uniformly strongly convex and to derive the rate of convergence of these potentials toward their unregularized limit. Our results generalize the results of (González-Sanz and Nutz, SIAM J.~Math.~Anal.) and (Wiesel and Xu, Ibid.) to the multivariate case and beyond the case of self-transport.
comment: 18 pages, no figures
☆ Fokker-Planck Analysis and Invariant Laws for a Continuous-Time Stochastic Model of Adam-Type Dynamics
We develop a continuous-time model for the long-term dynamics of adaptive stochastic optimization, focusing on bias-corrected Adam-type methods. Starting from a finite-sum setting, we identify a canonical scaling of learning rates, decay parameters, and gradient noise that yields a coupled, time-inhomogeneous stochastic differential equation for the parameters $x_t$, first-moment tracker $z_t$, and second-moment tracker $y_t$. Bias correction persists via explicit time-dependent coefficients, and the dynamics becomes asymptotically time-homogeneous. We analyze the associated Fokker-Planck equation and, under mild regularity and dissipativity assumptions on $f$, prove existence and uniqueness of invariant measures. Noise propagation is governed by $A(x)=\mathrm{Diag}(\nabla f(x))H_f(x)$. Hypoellipticity may fail on $\mathcal D_A\times\mathbb R^m\times(\mathbb R_+)^m$, where \[ \mathcal D_A=\{x\in\mathbb R^m:\exists j,\ e_j^\top A(x)=0\}\subset\{x:\det A(x)=0\}=\mathcal D_A^\dagger, \] and critical points of $f$ lie in $\mathcal D_A$. We show $\mathcal D_A^\dagger\neq\mathbb R^m$ and use this to prove exponential convergence of the Markov semigroup $μ_0P_t$ to a unique invariant measure, uniformly in $μ_0$. The proof uses a Harris-type argument, minorization on Lyapunov sublevel sets, control constructions, and hypoellipticity on $(\mathbb R^m\setminus\mathcal D_A)\times\mathbb R^m\times(\mathbb R_+)^m$. This provides a transparent continuous-time view of Adam-type dynamics.
☆ Symmetry and rigidity results for Serrin's overdetermined type problems in weighted Riemannian manifolds
We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of the ambient manifold. We further address the case of domains with non-compact closure for manifolds conformally equivalent to the Euclidean space, possibly degenerating or becoming singular at a point, where both the weight and the conformal factor are radial functions.
☆ Two-species system with nonlocal interactions driven by Riesz potentials
This paper investigates a system of nonlocal continuity equations modelling the interaction of two species coupled through Riesz-type potentials. The model incorporates self- and cross-interaction kernels of possibly different fractional orders. By exploiting optimal transportation theory and the theory of gradient flows in Wasserstein spaces, we establish the existence of weak solutions under singularity assumptions on all interaction potentials, provided the cross-interaction ones satisfy a symmetry condition. Our analysis extends previous results available for either single-species equations or multi-species systems with smoother cross-interaction kernels.
☆ Well-Posedness of the Helmholtz Equation with Rough Coefficients
We establish the well-posedness of the Helmholtz equation with rough and compactly supported coefficients in Rd under sharp regularity assumptions. Using a paraproduct calculus in rescaled weighted Besov spaces, we rigorously define the product between the solution and the coefficient at the lowest regularity level without renormalization. A rescaled Lippmann-Schwinger formulation is shown to be equivalent to the Helmholtz equation with the Sommerfeld radiation condition. We prove existence, uniqueness, and explicit wavenumber dependent resolvent estimates in a general Lp setting, including an L2 theory relevant to scattering amplitudes. The results provide a sharp analytic foundation for wave propagation and scattering in highly irregular media.
☆ Uniform-in-time diffusion approximations for multiscale stochastic systems
This paper establishes a quantitative, uniform-in-time diffusion approximation for the joint law of a broad class of fully coupled multiscale stochastic systems. We derive a precise characterization of the limiting joint distribution as a specific skew-product of the conditional equilibrium of the fast process and the homogenized law of the slow component, thereby providing a rigorous uniform-in-time formulation of the adiabatic elimination principle. The convergence rate explicitly separates the initial relaxation of the fast dynamics from the long-time homogenized evolution and depends only on the regularity of the coefficients in the slow variable. As a consequence, we obtain the first quantitative identification of the limiting stationary distribution of the original multiscale system and prove the commutativity of the limits $\eps\to0$ and $t\to\infty$ for a large class of observables. Our framework accommodates unbounded and irregular coefficients, degenerate structures, and weakly mixing dynamics. We illustrate its scope with three applications: {\it (i)} a uniform-in-time averaging principle for fast-slow systems; {\it (ii)} a uniform Smoluchowski--Kramers approximation for degenerate Langevin systems, yielding convergence of the joint position-scaled velocity law and global-in-time asymptotics of key thermodynamic functionals (e.g., total energy, entropy production, free energy); and {\it (iii)} the first uniform-in-time periodic homogenization result for SDEs with distributional drifts.
comment: 51pages
☆ Derivative estimates for SDEs with singular and unbounded coefficients
We develop a unified PDE-probabilistic framework for pointwise gradient and Hessian estimates of Markov semigroups associated with stochastic differential equations with singular and unbounded coefficients. Under mild local structural assumptions on the diffusion matrix and integrability/regularity conditions on the drift, we obtain quantitative sharp short-time regularization estimates as well as long-time decay bounds (including exponential and polynomial rates) for the first and second spatial derivatives of the semigroup. A distinctive feature of our results is the explicit dependence of these estimates on local norms of the coefficients (through scale-invariant quantities), without requiring any global smoothness, boundedness or uniform ellipticity. In particular, our approach allows for degenerate or highly irregular behavior at infinity, subject to suitable local ellipticity and Lyapunov/ergodicity controls. As applications, we establish solvability and regularity results for Poisson equations on the whole space with singular coefficients, and we derive pointwise gradient estimates for SDEs with distributional drifts via a Zvonkin-type transform.
comment: 38pages
☆ Weak-Strong Uniqueness for Second-Order Mean-Field Games
We extend the weak-strong uniqueness principle for mean-field game (MFG) systems to a broad class of second-order stationary and time-dependent problems. Under standard monotonicity, growth, and coercivity assumptions on the Hamiltonian, and relying strictly on the integrability exponents guaranteed by the existing theory for monotone MFG systems, we show that any weak solution must coincide with a given strong solution. Our analysis covers models with spatially dependent scalar diffusion coefficients, using monotonicity arguments and a coefficient-adapted mollification strategy to manage the variable diffusion terms. We extend this strategy to establish weak-strong uniqueness in the corresponding second-order, initial-terminal, time-dependent setting.
Finally, to address the critical quadratic growth regime, we derive a new a priori second-order estimate for a stationary MFG system with logarithmic coupling. This estimate provides quantitative bounds on the solution in terms of the data, and yields weak-strong uniqueness in the range where the improved integrability yields $L^2$ control of the density.
Since numerical and approximation methods for MFGs naturally yield weak solutions in the monotonicity sense, whereas strong solutions are known to exist in many settings, our results identify any weak limit produced by such methods with the strong solution whenever one exists.
☆ Discrete adjoint gradient computation for multiclass traffic flow models on road networks
This paper applies a discrete adjoint gradient computation method for a multi-class traffic flow model on road networks. Vehicle classes are characterized by their specific velocity functions, which depend on the total traffic density, resulting in a coupled hyperbolic system of conservation laws. The system is discretized using a Godunov-type finite volume scheme based on demand and supply functions, extended to handle complex junction coupling conditions -- such as merges and diverges -- and boundary conditions with buffer lengths to account for congestion spillback. The optimization of different travel-related performance metrics, including total travel time and total travel distance, is formulated as a constrained minimization problem and is accomplished through the use of an adjoint gradient approach, allowing for an efficient computation of sensitivities with respect to the chosen time-dependent control variables. Numerical simulations on a sample network demonstrate the efficiency of the proposed framework, particularly as the number of control parameters increases. This approach provides a robust and computationally efficient solution, making it suitable for large-scale traffic network optimization.
comment: 33 pages, 24 figures
☆ Li-Yau and Harnack estimates for nonlocal diffusion problems
These notes give a brief introduction to differential Harnack inequalities and summarise the main results of the mini-course ``Li-Yau and Harnack estimates for nonlocal diffusion problems'', presented by the author at the Seasonal School on PDEs ``Oscillation Phenomena, PDEs, and Applications: A Comprehensive School in Mathematical Analysis'', held at Ghent University in October 2025.
comment: 11 pages
☆ Verifying Well-Posedness of Linear PDEs using Convex Optimization
Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer-Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE -- a proper subspace of $L_{2}$ -- which can make them difficult to verify in practice. In this paper, we show how the Lumer-Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space $L_{2}$ and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer-Phillips conditions as operator inequalities on $L_{2}$. We show how these inequalities can be tested using convex optimization methods, establishing a least upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs.
☆ Fundamental solution and diffusion limits for the heat equation in a half-space with a diffusive dynamical boundary condition
We derive an explicit representation of the fundamental solution to the heat equation in a half-space of ${\mathbb R}^N$ with a diffusive dynamical boundary condition, and establish sharp pointwise upper and lower bounds. We also investigate qualitative properties of the associated solutions, including precise decay estimates. Furthermore, we analyze the diffusion limits of solutions to the initial--boundary value problem, and reveal the role of the diffusive dynamical boundary condition in the behavior of solutions.
☆ Trap behaviors for Brownian motions
This paper investigates the relationship between the geometric properties of a domain and the diffusion dynamics of Brownian motion, with a specific focus on the phenomenon of "trapping" in terms of the behavior of stochastic processes.
☆ Symmetric hyperbolic Schrödinger equations on tori
In this paper, we study the symmetric hyperbolic Schrödinger equations in the periodic setting. First, we prove full range Strichartz estimates on general tori by adapting Bourgain's major arc method. The result is sharp for rational tori. Second, on two-dimensional rational tori, we establish optimal local well-posedness for two hyperbolic nonlinear Schrödinger (HNLS) equations: the septic HNLS and the hyperbolic-elliptic Davey-Stewartson system.
comment: 23 pages
☆ On a Keller--Segel System with Density-Suppressed Motility, Indirect Signal Production, and External Sources
This paper investigates an initial-Neumann boundary value problem for a Keller--Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal production, captures a self-trapping effect suppressing cellular movement at high densities. We establish the global existence of classical solutions in arbitrary spatial dimensions for a broad class of non-increasing motility functions, both with and without external source terms. Furthermore, we demonstrate that any external damping source exhibiting superlinear growth ensures uniform-in-time boundedness. Conversely, in the absence of such damping, solutions may become unbounded as time tends to infinity. More precisely, in the two-dimensional homogeneous case with the exponentially decaying motility function $γ(v) = e^{-v}$, a critical mass phenomenon emerges: classical solutions remain uniformly bounded for subcritical initial mass, while supercritical initial masses can lead to infinite-time blow-up. Our analysis relies on the construction of carefully designed auxiliary functions along with refined comparison methods and iteration arguments.
☆ Existence of free boundaries for overdetermined value problems: Sharp conditions, regularity, and physical applications
This paper provides necessary and sufficient conditions for the existence of free boundaries in overdetermined value-problems (ODVP) for the Laplacian, and sufficient conditions for the bi-Laplacian, when the overdetermined boundary condition is non-constant. Using classical integral inequalities (Cauchy-Schwarz, Hölder, Hardy, eigenvalue bounds, Pohozaev and Reilly identities), we derive existence results for a broad class of free boundary problems arising in potential theory, plate theory, electromagnetism, and shape optimization. A regularity result for minimizers in the \(C\)-GNP class is established using the thickness function and the Wiener criterion, based on the geometric description of cusp points given in \cite{barkatou2002}. New results include refined estimates via interpolation inequalities, stability under perturbations, and connections with isoperimetric inequalities. Detailed analysis of the necessary and sufficient conditions for the existence of solutions to the ODVP is given. The physical interpretation of the bi-Laplacian problem \(\mathcal{B}(f,g)\) in the Kirchhoff-Love theory of thin plates is emphasized.
☆ Existence of Complementary and Variational Weak Solutions to Obstacle Problems for a Quasilinear Wave Equation
We prove the existence of weak solutions for the one obstacle problem associated with a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, and we also address the two obstacles problem. In contrast with the linear setting, for both strictly quasilinear cases we obtain continuous solutions in a weak complementary sense, which moreover satisfy a weak entropy condition in the free region where the string is not in contact with the obstacles. We further show that, in both the one and two obstacle cases, these solutions are variational solutions in a hyperbolic sense without the viscosity term.
comment: 32 pages
♻ ☆ The Linearized Floer Equation in a Chart
In this article, we are considering the Hessian of the area functional in a non-Darboux chart. This does not seem to have been considered before and leads to an interesting new mathematical structure which we introduce in this article and refer to as almost extendable weak Hessian field. Our main result is a Fredholm theorem for Robbin-Salamon operatorsassociated to non-continuous Hessians which we prove by taking advantage of this new structure.
♻ ☆ Analysis of a class of recursive distributional equations including the resistance of the series-parallel graph
This paper analyzes a class of recursive distributional equations (RDE's) proposed by Gurel-Gurevich [17] and involving a bias parameter $p$, which includes the logarithm of the resistance of the series-parallel graph. A discrete-time evolution equation resembling a quasilinear Fisher-KPP equation is derived to describe the CDF's of solutions. When the bias parameter $p = \frac{1}{2}$, this equation is shown to have a PDE scaling limit, from which distributional limit theorems for the RDE are derived. Applied to the series-parallel graph, the results imply that $N^{-1/3} \log R^{(N)}$ has a nondegenerate limit when $p = \frac{1}{2}$, as conjectured by Addario-Berry, Cairns, Devroye, Kerriou, and Mitchell [1].
comment: Submitted version. Minor edits from v2
♻ ☆ Removing small wavenumber constraints in Side B of the Probe Method
The Probe Method is an analytical reconstruction scheme for inverse obstacle problems utilizing the Dirichlet-to-Neumann map associated with the governing partial differential equation. It consists of two distinct parts: Side A and Side B. Both are based on the indicator sequence which is calculated from the Dirichlet-to-Neumann map acting on "needle-like" specialized solution of the governing equation for the background medium, whose energy is concentrated on an arbitrary given needle inside. In Side A, the limit of the indicator sequence-referred to as the indicator function-is computed before the needles touch the obstacle, and the boundary is identified as the point where this function first blows up. In contrast, Side B states the blow-up of the indicator sequence after the needles have come into contact with the obstacle. For the Helmholtz equation, the validity of Side B has long required a small wavenumber constraint. This paper finally removes this long-standing restriction, establishing the method's applicability for broader cases.
comment: 12 pages, typo corrected, Def.3.1 and Section 4 edited
♻ ☆ On the well-posedness of the KP-I equation
We revisit the local well-posedness for the KP-I equation. We obtain unconditional local well-posedness in $H^{s,0}({\mathbb R}^2)$ for $s>3/4$ and unconditional global well-posedness in the energy space. We also prove the global existence of perturbations with finite energy of non decaying smooth global solutions.
comment: We improved the manuscript. In particular, we corrected an error in the definition 2.2 of the admissible pairs for the Strichartz estimates
♻ ☆ Large-Time Analysis of the Langevin Dynamics for Energies Fulfilling Polyak-Łojasiewicz Conditions
In this work, we take a step towards understanding overdamped Langevin dynamics for the minimization of a general class of objective functions $\mathcal{L}$. We establish well-posedness and regularity of the law $ρ_t$ of the process through novel a priori estimates, and, very importantly, we characterize the large-time behavior of $ρ_t$ under truly minimal assumptions on $\mathcal{L}$. In the case of integrable Gibbs density, the law converges to the normalized Gibbs measure. In the non-integrable case, we prove that the law diffuses. The rate of convergence is $\mathcal{O}(1/t)$. Under a Polyak-Lojasiewicz (PL) condition on $\mathcal{L}$, we also derive sharp exponential contractivity results toward the set of global minimizers. Combining these results we provide the first systematic convergence analysis of Langevin dynamics under PL conditions in non-integrable Gibbs settings: a first phase of exponential in time contraction toward the set of minimizers and then a large-time exploration over it with rate $\mathcal{O}(1/t)$.
comment: 27pages
♻ ☆ The Boussinesq system in 3-dimensional bounded rough domains: Well-posedness in critical spaces and long-time behavior
We study the three-dimensional Boussinesq system in bounded rough domains, including bounded Lipschitz and $\mathrm{C}^{1,α}$ domains, within a critical functional framework. We establish existence and uniqueness results that are global in time for small initial data and local in time for arbitrary initial data. Well-posedness in critical endpoint Besov spaces with third index equal to $\infty$ is obtained in domains with Hölder continuous boundaries, relying on $\mathrm{L}^2$-maximal regularity in time. We also prove well-posedness in critical Besov spaces with third index equal to $1$, using $\mathrm{L}^1$-maximal regularity. In this $\mathrm{L}^1$-in-time setting, the analysis applies to arbitrary bounded Lipschitz domains. In any case, we show that the fluid velocity stabilizes exponentially for large times and that the temperature converges to the initial averaged temperature of the fluid. The linear theory -- fitting the adapted product estimates and vice versa -- is properly established prior to the nonlinear analysis. With this fully prepared linear framework in hand, the nonlinear estimates that follow are then handled in the critical framework with a simplified treatment -- especially in the case where the fluid velocity and the temperature belong to slightly larger spaces than $\mathrm{L}^2(\mathrm{W}^{1,3})$ and $\mathrm{L}^2(\mathrm{L}^{3/2})$ respectively -- when compared with previously known similar results in smooth domains. This approach relies on a robust linear theory and sharp product estimates based on operator-theoretic methods and Besov space techniques. Finally, as part of the analysis, we establish several new results for the underlying linear operators, including refined characterizations for the domains of fractional powers of the Neumann Laplacian and of the Stokes operator in bounded Lipschitz domains.
comment: A mistake in Prop. A.3 in v1 necessitated using Besov spaces with third index $\infty$ throughout the general framework instead of Lebesgue spaces. Proofs of bilinear estimates and most of the text remain unchanged. The uniqueness proof (Thm. 3.7), several intermediate results (e.g., Prop. 2.5), and the overall presentation were updated accordingly. Several typos and grammar errors were fixed
♻ ☆ Global strong solutions to the compressible Navier--Stokes--Coriolis system for large data
We consider the compressible Navier--Stokes system with the Coriolis force on the $3$D whole space. In this model, the Coriolis force causes the linearized solution to behave like a $4$th order dissipative semigroup $\{ e^{-tΔ^2} \}_{t>0}$ with slower time decay rates than the heat kernel, which creates difficulties in nonlinear estimates in the low-frequency part and prevents us from constructing the global strong solutions by following the classical method. On account of this circumstance, the existence of unique global strong solutions has been open even in the classical Matsumura--Nishida framework. In this paper, we overcome the aforementioned difficulties and succeed in constructing a unique global strong solution in the framework of scaling critical Besov spaces. Furthermore, our result also shows that the global solution is constructed for arbitrarily large initial data provided that the speed of the rotation is high and the Mach number is low enough by focusing on the dispersive effect due to the mixture of the Coriolis force and acoustic wave.
comment: This paper is published from Archive for Rational Mechanics and Analysis
♻ ☆ Positive weak solutions of a double-phase variable exponent problem with a fractional-Hardy-type singular potential and superlinear nonlinearity
In the present paper, we study a double-phase variable exponent problem which is set up within a variational framework including a singular potential of fractional-Hardy-type. We employ the Mountain-Pass theorem and the strong minimum principle to obtain the existence of at least one nontrivial positive weak solution.
♻ ☆ An analysis of the 2-D isentropic Euler Equations for a generalized polytropic gas law
In this paper we developed an analysis of the compressible, isentropic Euler equations in two spatial dimensions for a generalized polytropic gas law. The main focus is rotational flows in the subsonic regimes, described through the framework of the Euler equations expressed in self-similar variables and pseudo-velocities. A Bernoulli type equation is derived, serving as a cornerstone for establising a self-similar system tailored to rotational flows. We also developed an Ellipticity Principle for generalized polytropic gases, which is applied twice in this paper. To the best of the authors' knowledge, both applications appear for the first time. In particular, the analysis of the potential flow problem in the pseudo-subsonic regime is nontrivial for generalized polytropic gases when gamma< 1. In this setting, refined techniques, such as the Moser iteration method combined with suitable a priori estimates, are required. In the final section, the study extends to an analysis of a perturbed model, introducing the concept of quasi-potential flows, offering insights into their behavior and implications.
comment: 59 pages