Papers

This paper investigates the inverse problem of determining the spatially dependent damping coefficient in a time-fractional damped wave equation, where the damping term is given by a Caputo derivative of order \(\alpha\in(0,1)\). We first prove the well-posedness of the direct problem in exponentially weighted Sobolev spaces. Then, by means of the Fourier–Laplace transform in time, the nonstationary problem is reduced to a family of stationary elliptic equations with complex frequencies. Based on complex geometrical optics solutions and a suitable integral identity, we estimate the Fourier transform of the difference of two damping coefficients in terms of the difference of their Dirichlet-to-Neumann maps. Combining low-frequency estimates with a high-frequency decay argument, we obtain a conditional logarithmic stability result.

This article resolves some errors in the paper “Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance. Math. Ann. 392, 1051-1097 (2025)”. The errors are in the proof of contraction of a map associated with our equation in two and three dimensions.

This paper introduces a unified abstract variational and topological framework to characterize the spectral gap and eigenvalue distribution in high-contrast multiscale partial differential equations (PDEs). We rigorously prove that the exact location of the spectral gap is universally determined by the dimension of the local null space associated with the high-contrast inclusions. For systems with infinite-dimensional kernels, this location is strictly determined by the topological Betti numbers. Furthermore, we establish a multiscale Weyl's law via a spectral decoupling theorem, demonstrating that as the contrast approaches infinity, the multiscale spectrum bifurcates into two independent components: the Dirichlet spectrum of the background matrix and the internal Neumann spectrum of the inclusions. Using spectral homogenization theory, we also show that in the limit of vanishing periodicity, this expanding topological eigenspace asymptotically spans the entire spectral space of the macroscopic homogenized operator. These theoretical results are comprehensively verified through numerical experiments on diffusion, elasticity, fourth-order plate, Maxwell, and grad-div operators.

We investigate the Cauchy problem of three-dimensional compressible Navier-Stokes equations with far-field vacuum. Based on delicate energy estimates and structures of the systems under consideration, we show the global well-posedness and decay rates of strong solutions provided that some coefficient-coupled scaling invariant initial quantity is suitably small. In particular, our smallness conditions are independent of any initial data and known parameters in the systems. Moreover, there is no need to require the compatibility conditions on the initial data. Our results improve previous works.

We consider the two species (electrons and positive charged ions) Vlasov-Poisson system linearized around BGK waves. We formulate the problem as an equivalent Vlasov-Ampère system, that we write as a system of Schrödinger type in an appropriate Hilbert space, and with a selfadjoint Vlasov-Ampère operator as a Hamiltonian. We develop a complete stationary scattering theory. We identify the absolutely continuous spectrum and the singular spectrum of the Vlasov-Ampère operator, we construct the generalized Fourier maps, we prove that the wave operators exist, are complete, satisfy Birman's invariance principle, and that the stationary formulae hold. Using these results we prove that the BGK waves are asymptotically stable. We obtain a precise description of the large time behaviour of the solutions to the Vlasov-Ampère system Namely,for large times the phase-space densities of electrons and ions are asymptotic to the phase-space densities of solutions of the unperturbed Vlasov-Ampère system. This implies that they follow the trajectories of solutions to Newton's equations for electrons and ions with the potential of the BGK wave, in the sense that they are transported along these trajectories. Furthermore, we prove that Landau damping holds, that is to say, the electric field tends to zero in pointwise sense for large times.

We study refined asymptotics of scalar-flat ALE four-manifolds in the Tian–Viaclovsky setting, namely for self-dual or anti-self-dual metrics and for metrics with harmonic curvature. Starting from the ALE coordinates obtained by Tian–Viaclovsky, we construct preferred coordinates at infinity and identify the homogeneous $|x|^{-2}$ term in the metric expansion. This term splits canonically into a scalar part determined by the ALE ADM mass and an algebraic Weyl tensor at infinity. As an application, we consider scalar-flat Kähler ALE metrics on minimal resolutions $\pi:X\to\mathbb C^2/\Gamma$ of quotient surface singularities. In this case, the leading Weyl tensor at infinity vanishes exactly when the minimal resolution is crepant.

This paper proves weighted discrete Strichartz, or equivalently discrete restriction, estimates for two-dimensional exponential sums with fractional and perturbed polynomial phases. The new results cover three related classes of curves: the fractional Schrödinger-type curve $(n,n^{1+\nu})$, the fully fractional curve $(n^{1+\mu},n^{1+\nu})$ with $0<\mu<\nu$, and bounded perturbations of the polynomial curve $(n,n^k)$. In each case we obtain $\ell^2$-weighted $L^p$ estimates with decoupling-type dependence on the frequency scale, including both low- and high-$p$ regimes. The proofs are based on finite-type decoupling, together with localization and rescaling arguments adapted to curvature degeneracy, anisotropic homogeneity, and bounded perturbations of the phase. The introduction frames these estimates from the viewpoint of decoupling theory, with arithmetic results obtained by efficient congruencing used only as comparison points for the known cases.

In this paper, we study the 3-D incompressible fractional stationary MHD equations on the torus $\mathbb{T}^3$. For fractional power indices $\alpha_1,\alpha_2 >0,$ we prove that there exist infinite non-trivial stationary singular solutions of 3-D fractional MHD equations via convex integration. In particular, this result shows the non-uniqueness of weak solutions to the 3-D stationary MHD equations in some sub-critical Besov spaces.

We study small-noise asymptotics for a class of reversible stochastic evolution equations in infinite dimensions. The dynamics are of the form \[ dX_t=-A\nabla F(X_t)\,dt+\sqrt{2\beta^{-1}A}\,dW_t, \] where $F$ is a regular multi-well potential, $A$ is a selfadjoint mobility operator, $W$ is a cylindrical Brownian motion and $\beta\gg 1$ is the inverse noise strength. The invariant measure is a Gibbs perturbation of a Gaussian reference measure, and the resulting framework covers, in particular, the stochastic Allen-Cahn and stochastic Cahn-Hilliard equations on bounded intervals. In the double-well case, we derive a sharp asymptotic formula for the first nonzero eigenvalue of the generator. This gives an infinite-dimensional Eyring-Kramers law for the spectral gap, with exponential rate determined by the communication height and leading prefactor determined by the local quadratic behavior at the relevant minima and saddle points. Our approach provides a general strategy for lifting finite-dimensional Eyring-Kramers analysis to infinite-dimensional stochastic gradient systems.

Highly accurate stability boundary values for the ADER-DG method are obtained for arbitrary degrees $N$ of basis polynomials. In the linear case, stability is violated precisely when one of the matrix eigenvalues reaches $\lambda = -1$, regardless of the phase $\theta$. A rigorous mathematical framework for the stability is developed. The stability condition is significantly simplified, reducing it to the problem of calculating the roots of polynomials in the Courant number $\mathrm{CFL}$. The maximum of the Courant numbers $\mathrm{CFL}_{\rm max}(N)$ are calculated. These results are new and very convenient for practical use. A comparison of the obtained results with existing results reveals differences that may be significant for the selection of calculation parameters, especially for high degrees $N$. It is shown that widely used existing estimates $\mathrm{CFL}_{\rm max}(N) \propto 1/(2N+1)$ are overestimated. An interesting qualitative asymptotic $\mathrm{CFL}_{\rm max}(N) \propto 1/(N+1)^{2}$ is obtained. A rigorous direct proof of the approximation is presented. Approximation orders $p = N+1$ for arbitrary degrees $N$ are rigorously derived. A set of numerical experiments is carried out to apply the ADER-DG method to solving both a linear advection equation and an Euler system of equations. The results obtained in these calculations confirm the theoretical results well. In particular, an excess of the Courant number over the $\mathrm{CFL}_{\rm max}(N)$ by even 1% in the linear case immediately leads to significant instability of the numerical solution. The obtained estimates of the boundary Courant number in the nonlinear case are somewhat underestimated – by no more than 5%, which is due to the diffusivity and stability of the approximate Riemann solver. Empirical convergence orders are obtained, which are in good agreement with the theoretical results.

We investigate massive one-dimensional Dirac operators perturbed by diagonal matrix potentials of the form $i V$ where the function $V$ is real-valued and unbounded at infinity. For such operators we find an $L^2$-realization with non-empty resolvent set using generalized coercivity and Schur complement dominance techniques. In the prototypical Airy-Dirac case $V(x)=x$, $x \in \mathbb{R}$, we derive the precise asymptotic behavior of the resolvent as the spectral parameter tends to infinity and also as the mass $m$ tends to $0$. Finally, we find the asymptotics of the resolvent norm for general potentials $V$ in terms of the Airy-Dirac resolvent, which in particular yields an asymptotic shape of $\varepsilon$-pseudospectral curves and establishes the optimality of the pseudospectral region found in [32].

The $\sigma_k$-Loewner–Nirenberg problem is a fully nonlinear generalization of the classical Loewner–Nirenberg problem of constructing complete conformal metrics with constant negative scalar curvature in the interior of domains. For the $\sigma_k$-version, once $k > 1$, one must impose a negative admissibility condition to ensure ellipticity of the equation. In this note, we exhibit topological obstructions to admissibility and illustrate this with examples.

The pressureless Euler equations in dimension 1 describe the evolution of a distribution of mass over the real line. We are interested in sticky particles solutions: distributional solutions in which mass that collides sticks together for all time. Given initial data, a projection formula is known to give a sticky particles solution. We obtain a necessary and sufficient condition for this solution to go to equilibrium based only on the initial condition. Furthermore, we give a precise characterization of the equilibrium, as well as a formula for the time of collapse.

In the first version of Chang, Slote, Volberg, and Zhang's paper \cite{BSA_of_PTF}, the authors modify a nice recursive approach due to Kane in \cite{Correct_exponent_for_AS} where he bounded the average sensitivity of polynomial threshold functions. In \cite{BSA_of_PTF} Kane's argument was adopted to estimate the boolean surface area of polynomial threshold function. The bridge is a combinatorial averaging lemma considering all balanced partitions. The lemma serves as a substitute for an additive property of average sensitivity. With the lemma, one can apply a Kane-type algorithm to derive a recurrence. Solving the recurrence then gives an upper bound of $e^{C_d \sqrt{\log n}}$ for the boolean surface area. In the second version of the same paper, the authors derive a polylog upper bound for BSA of PTFs. The difference is that they use a tail estimate for the sensitivity function. With the help of a polynomial restriction lemma in \cite{poly_restriction} they sharpen the upper bound. It is noteworthy that when applying the polynomial restriction, each coordinate is put into each part independently with equal probability. As a result, a partition does not necessarily have equal-size blocks. In other words, it may not be balanced. In this note, we first investigate the effect of different partitioning. Second, we use the recursive method in the first version to derive a polylog upper bound for $\mathbb E[s(x)^{\eta}]$ where $\eta < 1/2$. It is interesting to note the phase transition that happens at $\eta=1/2$ in both versions of the proof (but in a completely different form). Section 2.5 treats that.

We prove uniqueness for a finite-dimensional anisotropic Calderón problem in dimensions $n\geq 3$. The unknown conductivity is a symmetric uniformly elliptic matrix field whose restriction to each cell of a known finite partition is polynomial; jumps across cell interfaces are allowed. We assume that the cells can be ordered so that each successive cell is accessible through sufficiently many flat faces. Under this assumption, the local Dirichlet-to-Neumann map on an exterior boundary patch determines every polynomial piece. The proof first recovers one polynomial matrix from its boundary invariants on several flat faces and then propagates the required local data through the recovered cells by unique continuation and Runge approximation. A general finite-dimensional analytic stability theorem also gives Hölder stability on compact admissible parameter sets.

We consider a one-parameter family of nonlinear wave equations on the $d$-dimensional torus, with polynomial nonlinearities of arbitrary degree $q+1$, where $q\geq 1$. We investigate the long-time behavior of high Sobolev $H^s$-norms of solutions in different settings. In the one-dimensional case, and for almost any value of the mass parameter $\mathtt{m}>0$, we prove exponentially long stability times for small initial data. The proof relies on normal form techniques together with suitable weak Diophantine conditions. In higher space dimensions, for initial data $u_0\in H^{s}$, $s \geq s_1 + 1$, satisfying suitable smallness conditions on the low Sobolev norm $H^{s_1}$ and on the $L^2$-norm, we prove a polynomial upper bound on the possible growth of the high Sobolev $H^{s}$-norm, over finite but exponentially long time scales in the regularity parameter $s_1$. The key ingredient consists in establishing suitable a priori tame estimates for the solution. The result applies in any space dimension $d\geq 1$ and for all values of the mass parameter $\mathtt{m}\geq 0$.

In this paper, we investigate the existence and qualitative properties of ground state solutions for the nonlocal Born-Infeld-Choquard problem \begin{equation*} \begin{cases} -div\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+ \omega u=\big(I_\alpha\ast |u|^{p}\big)|u|^{p-2}u, & \hbox{in }\mathbb{R}^N,\; N\geq 3, \\[5mm] u(x)\to 0, &\hbox{as }|x|\to +\infty. \end{cases} \end{equation*} where $p>\frac{N+\alpha}{N}$, $\omega=0,1$ and $0<\alpha<N$. The equation is driven by the mean curvature operator in Lorentz-Minkowski space, motivated by the Born-Infeld nonlinear electromagnetic theory, and is coupled with a Choquard-type nonlocal nonlinearity. Due to the inherent relativistic gradient constraint $|\nabla u| \le 1$, the associated energy functional lacks standard $\mathcal{C}^1$ regularity, preventing the direct use of classical variational techniques. We employ a non-smooth critical point theory on appropriate Pohožaev-type manifold to establish the existence of ground state solutions. Such non-smooth critical point theorem is abstract and we further show that it can be employed for strongly indefinite problem as well. We also demonstrate that these solutions are radially symmetric, and monotonously decay to zero at infinity.

In this note, we consider several quasiliniear minimization problems involving various Hardy type potentials for functions in $W^{1,p}(\Omega)$ with weighted mean zero. We prove that the strict inequality for the infimum yields the existence of a minimizer.

This paper establishes novel rigidity estimates for hyperbolic shells (surfaces with negative Gaussian curvature) and applies them to derive the \(\Gamma\)-limit of thin elastic shells. We prove a nonlinear rigidity estimate for \(H^1\) deformations on the mid-surface, and a nonlinear rigidity estimate for hyperbolic shells with clamped lateral boundary. The latter yields the optimal exponent \(h^{-4/3}\). As the main application, we characterize the \(\Gamma\)-limit of the nonlinear elastic energy for clamped hyperbolic shells across all scaling regimes \(\beta \in [0,2) \cup (8/3,\infty)\).

This article addresses the problem of proving the nonlinear instability of an equilibrium starting from its linear instability when there is no existence theory for the corresponding equations. We design a general method based on the use of analytic functions to overcome this difficulty and apply it to the classical problem of the stability of a plasma column in magneto-hydrodynamics (MHD) as an illustration.

We study the geometric regularization of positive closed currents by the Kähler-Ricci flow on compact Kähler manifolds. In a previous work of ours, it was shown that the Kähler-Ricci flow immediately smoothes out such a current when it has zero Lelong numbers. We study here the case when $T_0$ has divisorial singularities, showing that the flow gradually replaces the latter by Poincaré type ones, providing an approximation of $T_0$ by complete Kähler metrics with bounded curvature in a Zariski open set.

This paper investigates the rigidity of strain tensors on surfaces with sign-changing Gaussian curvature (mixed-type surfaces) and applies the results to determine the optimal thickness exponent in the first Korn inequality for thin shells. Using tools from Riemannian geometry and generalized tensor analysis, we derive an infinitesimal rigidity lemma for strain tensors, which establishes \(L^2\) regularity estimates for displacements decomposed into tangential and normal components. Specifically, we show that for a mixed-type shell with a middle surface \(S = S^+ \cup \Gamma_0 \cup S^-\) (where \(S^+\), \(S^-\) have positive and negative curvature, respectively, and \(\Gamma_0\) is a parabolic interface), the optimal constant in Korn's inequality scales as \(h^{4/3}\), matching the behavior previously established for hyperbolic shells. This result is obtained via a combination of geometric decomposition, Fredholm theory for linear operators, and compactness arguments to handle the curvature transition across \(\Gamma_0\). The findings bridge the gap between elliptic and hyperbolic shell theories, providing a unified framework for understanding rigidity in complex geometries with mixed curvature. The derived estimates are shown to be sharp, offering critical insights for the mechanical design of thin-walled structures with non-uniform curvature.

We classify any entire area-minimizing surface $M^2\subset\mathbb{R}^n$ with density $2$ at infinity as either planar, or the zero set of a quadratic holomorphic polynomial inside some affine copy of $\mathbb{C}^2$. We also prove algebraicity of $M^2$ when the tangent cone at infinity has multiplicity one with general density. In our proof we introduce a sheeting-type theorem for minimizing surfaces near a multiplicity-two plane at infinity, and a new proof of holomorphicity (different from \cite{micallef}) based on a general “calibrated at infinity” principle.

We consider a vectorial energy functional proposed in the physical literature as a simplified model for thin films of ferronematics – composite materials formed by dispersing magnetic nanoparticles into a nematic liquid crystals host. The model features two order parameters: a reduced $\Q$-tensor field $\Q$, describing the nematic liquid crystal component, and a vector field $\M$, accounting for the average polarisation vector field generated by the included particles. The energy contains a coupling term promoting alignment between $\Q$ and $\M$. It has been shown in~\cite{CDS1,CDS2} that, as a small parameter $\eps$ tends to zero, the energy of critical pairs $(\Q_\eps,\,\M_\eps)$ concentrates on distinct singular sets: a finite set of points for the $\Q_\eps$-component and the support of a $\H^1$-rectifiable varifold for the $\M_\eps$-component, with first variation supported on the singular set for the $\Q_\eps$-component. We show in this paper that, if the coupling constant is above a certain (explicit) threshold, then, up to rescaling its density by a material constant, such a limiting varifold has integer multiplicity.

We study families of smooth, embedded, regular planar curves $ \alpha : \left [-1,1 \right ]\times \left [0,T \right )\to \mathbb{R}^{2}$ with generalised Neumann boundary conditions inside cones, satisfying three variants of the fourth-order nonlinear curve diffusion flow: (1) curve diffusion flow with a length penalisation, (2, 3) two forms of constrained curve diffusion flow with fixed length. We prove in cases (2) and (3) for cone angle less than $\pi$, if the initial curve has small oscillation of curvature and the initial curve is sufficiently far from the cone tip, then the solution exists for all time and converges exponentially in the $C^\infty$-topology to a circular arc with the same length as the initial curve. In case (1), a similar result holds under suitable rescaling. In all cases, the limiting arc is centred at the cone tip.

In this paper, we consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with a step-like initial data. Based on the Riemann-Hilbert problem associated with the mKdV equation, we derive the long-time asymptotic expansion of the solution to the defocusing mKdV equation in two transition regions using the nonlinear steepest descent method. It comes out that the leading term in the expansion is shown to match the corresponding background constants. The subleading term, however, decays at the order $\mathcal{O}(t^{-2/3})$, and its coefficient is derived from the associated Painlevé \uppercase\expandafter{\romannumeral34\relax} model.

In this paper, we consider nonlinear Schrödinger equation. Duyckaerts–Roudenko [Comm. Math. Phys. 334 (2015), no. 3, 1573–1615] investigated time behavior of solutions to the equation, whose mass-energy is greater than that of the ground state. We revisit the results and show by using the action. Moreover, we apply it to other equations without scale invariance.

For $s \in (1/4,1)$ and any degree the only $W^{s,\frac{1}{s}}$-minimizers for $\mathbb{S}^1 \to \mathbb{S}^1$ maps are Blaschke products. This gives a resolution of Open Problems 23 and 24 in Brezis-Mironescu's mappings to the circle book, as well as Brezis' Favorite Open Problem 5.4 in this $s$-range. Previous results of this type were partial and restricted only to a small neighborhood of $s=\frac{1}{2}$. In particular, Brezis' Favorite Open Problem 5.1 is completely settled. Moreover, as a consequence of the argument, one also obtains linearized stability results.

We revisit the critical Hardy–Rellich inequalities recently established by Castro in "A critical Hardy–Rellich inequality (preprint, arXiv:2511.16537, 2025)". Using the classical one-dimensional weighted Hardy inequality in Emden–Fowler variables, we prove that the weighted inequality holds exactly for $a\neq N$, with $a=N$ as the unique critical value. We derive explicit constants for $N\ge2$, obtain the sharp constant in dimension one, and extend the $\Delta u$ formulation to the Muckenhoupt range $-N<a<N(N-1)$, $a\neq N$. This provides a partial solution to an open problem raised in Castro's work.

We study the stability of transition fronts for the one-dimensional Cahn-Hilliard equation. More precisely, we prove that for any Hölder initial datum sufficiently close to a front in the $L^2$ norm, the corresponding solution of the Cahn-Hilliard equation exists globally and converges, up to a dynamical shift, to the front as time tends to infinity. The result holds for a class of double-well potentials in a $C^4$-neighborhood of the standard quartic potential. For the proof, we develop an $L^2$-stability framework adapted to Cahn-Hilliard fronts. The key ingredient is a nontrivial second-order Poincaré-type inequality that reveals a coercive structure in the indefinite quadratic energy form associated with the linearized operator about the front, once the dynamical shift is taken into account. This yields an $L^2$-contraction estimate and, via a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted $L^2$ topology. The stability analysis requires only the $L^2$-smallness of the initial perturbation; no higher-order smallness, spatial localization, or moment assumptions are imposed.