Papers

We study extreme wave formation for the Korteweg-de Vries equation on the torus with random initial data of average size $\epsilon$. We establish a large deviations principle for the supremum of the solution over arbitrarily long polynomial timescales $t \leq \epsilon^{-n}$ for any fixed natural number $n$. This identifies the leading-order asymptotics of the probability of observing unusually large amplitudes. In this integrable setting, the dynamics evolves on invariant tori where Fourier moduli are almost conserved, ruling out mechanisms for extreme wave formation based on resonant energy exchange. As a result, large amplitudes can only arise through coherent structures or dispersive focusing, which corresponds to the quasi-synchronization of many phases. We show that the latter is dominant in the weakly nonlinear regime. Our approach combines a Birkhoff normal form analysis with probabilistic arguments, exploiting the stability of the integrable dynamics to control the probability of phase quasi-synchronization over long timescales.

We establish existence and uniqueness of global, bounded weak solutions to quasilinear PDEs with bounded, uniformly continuous initial data and investigate their properties. Moreover, we establish existence of bounded weak solutions when the initial data is merely bounded.

We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and square-integrable, and which, in finite time, transition to $C^{1-\mu}$ regularity for $\mu \in [1/2,1)$ near the boundary, leading to the gradient blowup at the boundary. It is based on stability analysis of self-similar waiting time solutions \cite{JLN2025} recently constructed by the authors.

We study the Schrödinger evolution generated by the Pauli-Fierz Hamiltonian, a model for nonrelativistic quantum electrodynamics, in the classical limit $\hbar \rightarrow 0$. In this regime, we rigorously derive the Newton-Maxwell equations of classical electrodynamics as effective dynamics approximating the time evolution. Our result complements prior work by an alternative derivation that provides explicit estimates on the rate of convergence, justifying the validity of the approximation for a special class of initial data.

We develop a structure-preserving solution framework for the optimal control of the time-dependent Maxwell's equations. Building on a well-posedness theory for a weak form of the forward problem, we first analyze a forward solver that couples Nédélec and Raviart–Thomas finite elements with Crank–Nicolson time stepping. The solver preserves the de~Rham structure, enforces a discrete Gauss law, exactly satisfies a per-time-step energy balance, and converges to the weak solution under low regularity assumptions on the problem data, which are dictated by the optimal control setting. To control the Maxwell system, we add the curl of a space-time current density as a source to Ampére's law. The curl form yields charge conservation without auxiliary constraints. We prove the well-posedness and continuity of the control-to-state map, derive the adjoint system and a gradient representation for a tracking-type objective functional, and formulate a discrete optimization scheme that inherits structure preservation from the forward solver. Our discrete stationarity conditions are consistent with their continuous counterparts, and the discrete optimal controls converge, with mesh and time refinements, to the continuous optima. We demonstrate the merits of our optimal control formulation and the theoretical developments by numerically solving a series of source-cloaking model problems.

Solutions to hyperbolic conservation laws can be approximated in many different ways: by vanishing viscosity, relaxations, discrete or semi-discrete numerical schemes, approximation with a nonlocal flux, etc$\ldots$ For some of these methods, general ${\bf L}^1$ convergence results are available. Aim of this paper is to understand the local behavior of these approximations, in a neighborhood of point where the hyperbolic solution has a singularity. Specifically: a point along a shock, or where two shocks interact, or where a new shock is formed. Given a sequence of $\epsilon$-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as $\epsilon\to 0$ a well defined limit is obtained. This corresponds to a specific “eternal solution" (globally defined both in space and in time) to the approximating equation. Precise results this direction are here given, in the case of vanishing viscosity.

We consider nonnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel system \begin{align*} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\Delta u-\nabla \cdot \big(u\nabla v\big),\\ &0=\Delta v -\mu + u , \\ \end{array}\right. \end{align*} where $\mu$ is the spatial average of $u$, under homogeneous Neumann boundary conditions in a ball in $\mathbb R^n$ for $n\geq 3$. In two dimensions, it is well established that solutions blowing up in finite time converge to a Dirac profile in the vague topology. In contrast, for $n\geq 3$, blow-up solutions with finite existence time do not appear to exhibit such concentration behavior. By generalizing to measure-valued solutions corresponding to accumulated densities of $u$, we extend the analysis beyond the blow-up time. Within this framework, we establish the existence of a minimal solution \[ u(t)=\theta(t)\delta_0 + \rho(\cdot,t) dx, \qquad t \geq 0, \] where $\rho$ is integrable and $\theta$ is increasing and right-continuous. We further construct a class of initial data for which $\theta(t_0)>0$ for some $t_0>0$, thereby establishing the formation of a Dirac mass at the origin. Unlike in the case $n=2$, the singular mass does not jump to a positive level instantaneously; instead, $\theta$ becomes positive continuously. Moreover, $\theta$ is strictly increasing on $[t_0,\infty)$, and the entire mass is asymptotically absorbed at the origin.

This paper investigates the existence and uniqueness of solutions for a nonlinear evolution equation governed by an m-accretive operator A in a Banach space, presenting a perturbation term that does not satisfy the Lipschitz condition.

In this paper, we study an inverse boundary value problem for the Jordan–Moore–Gibson–Thompson equation on a simple Riemannian manifold. We consider an all boundary measurement map that maps Dirichlet boundary data and initial data to the corresponding Neumann-type boundary data and final-time data. Our main result shows that the nonlinear acoustic coefficient $\beta$ is uniquely determined by this measurement map, and the linear damping coefficients $\alpha$ and $q$, along with the internal source term $F$, can be recovered up to a gauge symmetry. As a corollary, we also establish a specific case in which all coefficients are uniquely recovered. The proof relies on the method of first-order and second-order linearization and on the construction of geometric optics solutions. In the intermediate step, we establish the unique recovery of the lower-order coefficients in the linearized MGT equation.

Viscoelastic rate-type fluid models constitute a fundamental framework for the mathematical description of complex materials exhibiting coupled elastic and viscous effects, with a wide range of applications in engineering, biomaterials, and medicine. In realistic regimes, thermal effects are essential and lead to strongly coupled systems in which heat conduction and temperature-dependent constitutive laws play a decisive role. In this paper, we develop a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids. We establish the existence of a global weak solution in the full three-dimensional setting. In contrast to the existing literature, no smallness, regularity, or structural restrictions on the initial data are imposed beyond natural energy and entropy bounds, and no additional regularising mechanisms such as artificial stress diffusion are required. The analysis is based on a weak-strong framework combining energy and entropy balances with compactness tools, allowing us to treat the full nonlinear coupling between the fluid velocity, the temperature, and the elastic stress.

We study the Liouville equation $\triangle u+e^{2u} =0$ in a Riemannian surface $(M, g)$ with nonnegative $Ricci$ curvature. Under some asymptotic lower bound assumptions, we classify all the solutions to this equation, meanwhile we obtain the rigidity results for the ambient manifold. Note that our assumptions are optimal in some sense and differ from the classical assumption of finite total curvature.

In this paper, we study the $L^p$-boundedness of Stein's square function $\mathfrak{S}^{\alpha}(\mathcal{L})$ associated with the sub-Laplacian $\mathcal{L}$ on Métivier group $G$. A key aspect of our result is that the smoothness condition is expressed in terms of the topological dimension $d$ of the underlying Métivier group $G$. Consequently, we also present several applications of the $L^p$-boundedness of $\mathfrak{S}^{\alpha}(\mathcal{L})$. First, we provide an alternate proof of the sharp $L^p$-boundedness result for spectral multipliers on Métivier groups, recently obtained by Niedorf [Niedorf, Studia Math., 2025]. Next we prove $L^p$-boundedness of maximal spectral multipliers and consequently establish sharp $L^p$-boundedness result for the maximal Bochner-Riesz operator on Métivier groups, which also yields pointwise almost everywhere convergence of Bochner-Riesz means with smoothness parameter given in terms of the topological dimension of $G$. In case of Métivier groups our result improves upon the existing works of Mauceri-Meda [Mauceri, Meda, Rev. Mat. Iberoam., 1990] and Horwich-Martini [Horwich, Martini, J. Lond. Math. Soc., 2021]. Our result further imply the mixed norm regularity estimates for the solution of fractional Schrödinger equation on Métivier groups, where the regularity index is again expressed in terms of the topological dimension of $G$. Finally, we study the $L^{p_1}(G) \times L^{p_2}(G)$ to $L^p(G)$ boundedness of the bilinear Bochner-Riesz means and its maximal version, associated with the sub-Laplacian on Métivier group $G$. Our result improves upon the recent work of the author with Bagchi and Molla [Bagchi, Molla, Singh, J. Funct. Anal., 2026] in the range $2\leq p_1, p_2 <\infty$. In the same range, ......

We consider a quasi-static nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin-Voigt rheology where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. We refine the discretization schemes in [Badal-Friedrich-Kružík '23, Mielke-Roubíček '20] by imposing frame indifference already at a time-discrete level. This is justified both analytically and numerically.

This paper investigates the heat equation on a bounded domain with a Robin boundary condition, where the reactivity parameter (or killing rate) is modeled as a continuous-time Markov chain. We analyze the system under two stochastic frameworks using a functional analytic approach. First, we examine the annealed case, which accounts for the joint stochasticity of the diffusion and the switching mechanism. We describe the solution via a strongly continuous contraction semigroup on a product space. We identify its infinitesimal generator, which incorporates the state-dependent Robin conditions into its domain, and provide a corresponding Feynman-Kac formula. Second, we study the quenched setting for fixed realizations of the switching paths. We characterize the solution through a non-autonomous evolution family (propagator) and derive a Feynman-Kac-type representation involving the boundary local time of a reflected Brownian motion. We prove an averaging principle in the fast-switching limit, showing that the system converges to a deterministic Robin problem. These results are applied to a biophysical model of stochastically gated receptors on cell membranes.

In this paper, we present a new approach to obtain so-called damping estimates for self-similar solutions to general hyperbolic relaxation systems. Such damping estimates are an important part of the stability theory of relaxation profiles with and without subshock where they enable the closure of nonlinear stability arguments. We extend the damping estimates obtained in Mascia and Zumbrun (2005) and Yang and Zumbrun (2020) from the $L^2$-case to the $L^\infty$-case and, at the same time, generalize the $L^2$-estimates to the general non-symmetric setting. Compared to previous deductions of damping estimates, the method chosen here is considerably simpler and avoids delicate Kawashima-type estimates (Mascia and Zumbrun (2005) and Yang and Zumbrun (2020)) and the use of pseudo-differential calculus (Zumbrun 2026). The novel nonlocalized damping estimates open the door to a general stability theory of shock profiles of hyperbolic relaxation systems under nonlocalized perturbations in one space dimension.

We introduce Gårding polynomials, a class of real multivariate polynomials characterized by positivity regions that are invariant under translation by positive vectors and closed under strictly positive affine transformations. We prove that this geometric formulation is equivalent both to a reduction to the multi-affine setting via polarization and to a recursive criterion in terms of partial derivatives. The class of Gårding polynomials strictly extends that of real stable polynomials while preserving many of their structural properties. In particular, multi-affine Gårding polynomials with nonnegative coefficients satisfy the Rayleigh property, and their positive univariate specializations have ultra log-concave coefficient sequences. The Gårding property for several matroid generating functions is preserved under natural matroid operations. As applications, we derive new negative dependence results for generating functions associated with various classes of matroids and graphs, including examples previously beyond the scope of real stability and Lorentzian methods. We further obtain analogous results for characteristic polynomials arising from certain matrix classes.

In this paper, we study the three-dimensional inviscid incompressible resistive Hall-MHD system in the axisymmetric setting with nontrivial swirl velocity and purely azimuthal magnetic. Assuming only that the swirl component of the initial velocity is sufficiently small, we prove that the lifespan of the strong solution can be sufficiently large. An explicit lifespan lower bound in terms of the size of the initial swirl is given. Moreover, we also study the behavior of the lifespan as the resistivity tends to zero.

This article develops a Hamilton–Jacobi theory for non-conservative classical field theories, with particular emphasis on dissipative systems, in the framework of co-oriented k-contact geometry. We introduce evolution k-contact k-vector fields, extending the contact evolution formalism to field theories, and analyse the corresponding Hamilton–De Donder–Weyl equations. Moreover, we develop two distinct families of Hamilton–Jacobi theories: a z-independent approach, based on the reconstruction of the dynamics from an integrable k-vector field defined on the base manifold of $(\bigoplus^kT^*Q)\times\mathbb{R}^k\to Q$, and a z-dependent approach, where the integrable k-vector field is defined on the base manifold of $(\bigoplus^kT^*Q)\times\mathbb{R}^k\to Q\times\mathbb{R}^k$. We develop in detail the important case of Hamiltonian functions with affine dependence on the dissipative variables, show how quadratic dependence on these variables can be used structurally to enlarge the range of applications, and recover the ordinary contact Hamilton–Jacobi theory as the particular case k=1, while removing some technical assumptions appearing in previous formulations. Our theory is illustrated through several representative examples, including the telegrapher/Klein–Gordon equation, a dissipative Hunter–Saxton equation, a simple dissipative non-regular first-order field model, and a relativistic thermodynamic model.

We construct pulse-type approximate solutions to nonlinear hyperbolic equations near diffractive points, allowing arbitrary (even infinite) order of grazing. We show that in low regularity spaces and the high frequency limit, such solutions can be approximated by a sum of incoming and reflected pulses constructed using incoming and reflected phases and profiles that satisfy transport equations. New low-regularity estimates comparing the size of pulses to the size of their profiles are required. Earlier geometric optics results for pulses assumed much higher regularity, and considered only propagation in free space or transversal reflection at boundaries.

We establish a half-space theorem à la Hoffman and Meeks for nonlocal minimal surfaces. Differently from the classical case, our result holds in every dimension.

\indent In this paper, we study a class of parabolic-elliptic Keller-Segel systems with diffusion sensitivity dependent on spatial position, given by type \begin{equation} \left\{ \begin{array}{ll} u_{t} = \bigtriangledown\cdot(|x|^{\beta} \bigtriangledown u)-\bigtriangledown\cdot(u^{\alpha} \bigtriangledown v), 0=\bigtriangleup v-\mu +u, \qquad \mu:=\frac{1}{|\Omega|}\int_{\Omega}udx,\end{array}\right. \end{equation} under homogeneous Neumann conditions in a ball $\Omega=B_{R}(0)\subset \mathbb{R}^{n}$ with $\alpha \ge 1$, $\beta>0$ and $n\ge 2$.\par \indent It is proved that any nonconstant nonnegative radial initial data $u_{0}\in C^{\theta}(\overline{\Omega})$, where $\theta \in (0,1)$, there exists a radially symmetric classical solution of the system (0.1) in $(\Omega \setminus \{ 0 \})\times (0,T)$ for some $T>0$; moreover, if the initial values $u_{0}\in C^{1+\theta}(\overline{\Omega})$ for some $\theta \in (0,1)$ and satisfy a certain compatibility criterion and are radially decreasing, then this solution is bounded and unique in $(\Omega \setminus \{ 0 \})\times (0,T^{*})$ with $T^{*}<T$.\par Finally, it is found that the initial mass corresponding to this parabolic-elliptic problem (0.1) is sufficiently concentrated to allow the solution to blow up in finite time.

We address the question of the large-time behavior of solutions to reaction-diffusion equations in periodic media. We start with the description of the asymptotic shape of the invasion set, which is characterized by the Freidlin-Gärtner formula. We outline a proof of the formula that holds true for general types of reaction terms. We then present some recent results, obtained in collaboration with H. Guo and F. Hamel, for (weakly) bistable equations. They include a regular version of the Freidlin-Gärtner formula and the convergence in profile towards pulsating traveling fronts for solutions with either bounded or unbounded initial support.

We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and \[ \left\lvert H(x,t,\nabla u)\right\rvert\leqslant g(x,t)\left\lvert \nabla u\right\rvert^{\delta(x)} \] with \[ \delta (x)=\frac{p(x)(N+1)-N}{(N+2)(p(x)-1)}(p^--1) \quad \text{and} \quad p^-=\underset{x\in\bar{\Omega}}{min}\,p(x). \] By virtue of truncation metheod,the monotone operator theory and a gradient estimate we prove existence of renormalized solutions without coercivity condition on lower order term in the framework of variable exponents.

In this paper we study when the origin $(0,0)$ is a regular (or irregular) boundary point for the so-called soda can domains of the type \[ \Theta_{l,\theta}:= \{(x,t) \in \mathbf{R}^{n+1}: 0<-t < \theta |x|^l <\theta\}, \quad \text{with $l,\theta >0$,} \] for the $p$-parabolic equation $\partial_t u- \Delta_p u=0$, where $1<p<\infty$. For $p<2n/(n+1)$ and for the heat equation (i.e.\ $p=2$) we completely determine when the origin is regular for soda can domains. The domains $\Theta_{l,\theta}$ have nonconvex time sections with power dependence on time. For domains with rotationally symmetric convex time sections with power dependence on time, the regularity of the origin as the last point was characterized by Petrovskii (in 1935) for the heat equation, and almost completely in the nonlinear case ($p \ne 2$) in our earlier paper (joint with Gianazza, Math. Ann. 368 (2017), 885–904).

We investigate the existence of positive solutions to fractional equations presenting a double criticality: a multi-polar Hardy-type potential and a Sobolev critical nonlinearity. The nonlocal nature of the operator and the absence of explicit ground states for the single-pole equation stand as major difficulties. We overcome these obstacles by passing to an extended formulation of the problem and by establishing sharp asymptotic estimates for the solutions in the case of a single pole. Then, through a concentration-compactness argument, we show that the existence of minimizers is dictated by the magnitude of the masses and the mutual distances between the corresponding poles.

The primary goal of this paper is to give a precise definition and prove existence and uniqueness of multiphase quadrature domains for subharmonic functions, ensuring that the prescribed measures are supported in the interior of the resulting domains. The approach to prove existence is based on a variational framework, where we minimize an energy functional over so called segregated states. In this respect we refine earlier results in this direction. But we also show that this approach alone is not enough for two reasons. First of all it seems hard to get existence results which ensure that the interior support condition is satisfied. And second it may happen, as we show by an example, that a multiphase quadrature domain exists but is not a minimizer of the energy functional. The main novelty of this work is the study of minimizers, if they exist, of the energy functional over a subset of the segregated states given by a natural constraint with respect to the given measures. From this approach we are able to prove uniqueness, and also give sufficient conditions for existence. We also give an example showing that, unlike the energy minimization and partial balayage approaches which are equivalent in the one-phase case, this equivalence breaks down already in the two-phase setting.

We consider the following two-component coupled nonlinear Schrödinger (CNLS) system: \[ \begin{cases} -\Delta u +(P(x) + \lambda ) u=\mu_1 u^3+\beta u v^2, & \text{in } \mathbb{R}^N,\\ -\Delta v +(Q(x) + \lambda ) v =\mu_2 v^3+\beta vu^2, & \text{in } \mathbb{R}^N \end{cases} \] with the mass constraint $\int_{\mathbb{R}^N} (u^2+v^2)\,dx = \rho^2$ for $N=2,3$, where $\rho>0$ is a parameter. By employing the Lyapunov-Schmidt reduction and local Pohozaev identities, we establish the existence and local uniqueness of normalized multi-peak solutions: the result holds for sufficiently small $\rho$ when $N=3$, and for $\rho$ approaching a critical threshold when $N=2$. The main difficulty lies in that the mass constraint involves interactions among all concentration points, while a more refined characterization of such normalized solutions further requires sharp order estimates. In this work, we have discovered some new phenomena that differ from those of solutions without mass constraint and single-peak solutions.

The accurate simulation of interface-dominated solid mechanics problems on complex microstructures remains challenging, particularly when interface-fitted quadrilateral or hexahedral meshes are difficult to generate. We extend the shifted boundary method (SBM) to cohesive-zone formulations and introduce the Shifted Cohesive Zone Method (SCZM), with applications to crystal plasticity on non-interface-fitted meshes. By shifting the enforcement of traction-separation laws from the true interface to a nearby surrogate interface, SCZM enables the use of standard finite element spaces while avoiding the meshing burden associated with interface-conformal discretizations. We present a simplified SCZM weak form defined on the surrogate interface, leading to a straightforward implementation of the nonlinear residual and consistent tangent matrix. The method is implemented in the open-source MOOSE framework and coupled with constitutive models from NEML2, enabling simulations with linear elasticity, multiple traction-separation laws, and history-dependent crystal plasticity. We further develop a geometry-aware, PCA-enhanced point classification algorithm to accelerate surrogate-domain construction. Verification and benchmark studies in two and three dimensions demonstrate that SCZM achieves first-order convergence for non-interface-fitted interface problems and closely matches interface-fitted reference solutions in terms of reaction forces, surface energy release, deformation, stress fields, and damage evolution. These results indicate that SCZM provides an accurate and efficient framework for modeling interface mechanics in complex microstructures without requiring interface-fitted meshes.

We consider a linearized inverse scattering problem for elastic waves. We prove that a fully anisotropic perturbation of the elastic parameters around an isotropic and homogeneous reference can be uniquely determined by (single-)scattered waves. We also give a quantitative stability estimate for an isotropic perturbation, and as a consequence a rigidity result is established.

This work establishes a Lipschitz stability result for identifying unknown polygonal inclusions along with their unknown constant conductivity values, given boundary measurements encoded in the Dirichlet-to-Neumann map. We also obtain Lipschitz estimates for the Hausdorff distance between inclusion boundaries and for the difference of the conductivity values