Papers

We study systems of reaction-diffusion equations perturbed by multiplicative noise, where the reaction terms satisfy quasipositivity, a triangular mass-control structure, and polynomial growth. Our results apply to a broad class of reaction-diffusion systems arising, most notably, in chemistry and biology. In the deterministic setting these assumptions are known to guarantee the global existence of solutions. In the stochastic setting, however, reaction-diffusion systems have typically been analyzed under different assumptions on the reactions that preclude many natural models, such as reversible chemical reaction networks modeled by the mass-action law, and the question of global existence and uniqueness under a mass-control structure has remained open. In this work, we show that stochastically perturbing reaction-diffusion systems with triangular control of mass by suitable multiplicative noise leads to solutions that exist for all time.

In this paper, we investigate the blow-up in finite time and the corresponding lifespan estimates for a weakly coupled system of wave equations on a compact Lie group. In particular, we show how the Cauchy data and the presence of lower order terms affect the lifespan of local in-time solutions.

We study complete non-compact manifolds of positive scalar curvature, with a focus on how curvature decay is constrained by topology at infinity. Our first main result shows that topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry. This result provides a conceptual generalization of recently discovered examples of metrics with quadratic scalar curvature decay. Building on this decay mechanism, we develop an obstruction theory localized at the ends of non-compact manifolds. Using $\mu$–bubble exhaustions together with the analysis of stable minimal hypersurfaces and index theory, we obtain qualitative obstructions to uniformly positive scalar curvature on individual ends.

In this paper, we establish the global well-posedness of the one-phase Muskat problem with surface tension for small initial data. This problem describes the motion of the interface separating a wet region from a dry region within a porous medium, a process governed by Darcy's law. Although physically essential, the inclusion of surface tension introduces an additional challenge. We prove that if the initial free boundary is sufficiently small in $H^s$, $s>d/2+1$, then the problem admits a unique global strong solution. Moreover, the solution converges to zero in Lipschitz norm as $t\rightarrow\infty$. To the best of our knowledge, this work constitutes the first global well-posedness result for the one-phase Muskat problem with surface tension.

In this paper, we study Vekua-type operators associated with diagonal operators on compact Lie groups. Characterizations of global hypoellipticity and global solvability properties are presented on classes of Vekua-type operators with constant coefficients. We also present sufficient conditions in order to get global solvability for a class of Vekua-type operators with non-constant coefficients.

The local H theorem is shown to hold for the Enskog equation with a modified Enskog factor proposed by the authors [Phys. Rev. E 111, 065108 (2025)]. This is a stronger statement than the global one in the same paper and has been obtained along the lines of Mareschal et al. [Phys. Rev. Lett. 52, 1169-1172 (1984)] for the modified (or revised) Enskog equation. Furthermore, it is shown that the local H theorem also holds for the corresponding Enskog-Vlasov equation.

We establish new Liouville-type theorems for the stationary Navier-Stokes equations in $\mathbb{R}^3$. A central open problem in this context is whether the classical $L^{9/2}(\mathbb{R}^3)$ condition of G.Galdi can be relaxed. In this note we show that this global integrability requirement can indeed be weakened. More precisely, we prove that triviality already follows under assumptions of the form $u \in L^{9/2 + \varepsilon(\cdot)}(\mathbb{R}^3)$, where $\varepsilon(\cdot)>0$. As a consequence, we obtain a localized Liouville theorem: it is sufficient to impose this integrability condition only at infinity, with no additional assumptions on the behavior of $u$ inside a compact set. This highlights that the mechanism enforcing triviality is purely asymptotic. Our approach relies on a general uniqueness result in the framework of Lebesgue spaces with variable exponents, which naturally captures the coexistence of different integrability regimes across the domain.

We study a stochastic branching model for a population structured by a quantitative phenotypic trait and subject to births, deaths, and mutations. In a regime of large population and small mutations, and in logarithmic scales of size and time, we derive a certain class of free boundary Hamilton-Jacobi equations with state constraints from the stochastic individual-based system. This goes beyond the classical Hamilton-Jacobi equations obtained from deterministic models by taking into account the possible extinction of the system in certain regions of the trait space. The proof is obtained by combining methods for the analysis of Hamilton-Jacobi equations with probabilistic tools from the theory of large deviations and branching processes.

This paper investigates the blow-up of solutions to scale-invariant semilinear wave equations featuring the damping term $\frac{\mu}{1+t} \partial_t u$, the mass term $\frac{\nu^2}{(1+t)^2} u$, and a time-derivative nonlinearity $| \partial_t u |^p$. The principal contribution of this work is the demonstration that the sign of the discriminant $\delta = (\mu-1)^2 - 4\nu^2$ is not a structural prerequisite for determining the blow-up range. Indeed, we show that even in the regime $\delta < 0$, the blow-up region remains invariant and is uniquely determined by the shifted dimension $n+\mu$, aligning with the Glassey-type critical exponent. Our result suggest that the classical restriction $\delta \ge 0$ is due to a technical tool rather than an intrinsic feature of the blow-up mechanism.

We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge–Ampère equation with $\mathcal{C}^\infty$-smooth data does not in general admit $\mathcal{C}^{1,1}$-smooth solutions. This working draft is a prelude to potential-theoretic solutions to some extension problems for mappings that were thought to rely on such $\mathcal{C}^{1,1}$-smooth solutions.

We consider a fluid-structure interaction problem involving a viscous, incompressible fluid flow, modeled by the 2D Navier-Stokes equations, through a thin deformable elastic tube, displacement of which is not known a priori. The elastodynamics problem is given by 1D plate equations. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic coupling conditions at the fluid-structure interface. The fluid flow is driven by dynamic pressure data imposed at the inlet and outlet of the tube. We impose the Navier-slip boundary condition at the deformable fluid-structure interface and at the bottom rigid boundary of the fluid domain. Hence, beyond the usual geometric nonlinearities arising from nonlinear coupling in FSI with no-slip, the analysis is more challenging due to the possibility of tangential jumps of the fluid and structural velocities at the moving interface. We first discuss the existence of weak solutions and then establish a `hidden' spatial regularity result for the structure displacement. Our main result proves the existence of a finite time for weak solutions at which the compliant upper boundary meets the lower boundary (i.e., the tube collapses), provided that there is a sufficient pressure drop across the channel. This resolves the ”no-collision” paradox identified by Grandmont and Hillairet in the no-slip setting in [Arch. Ration. Mech. Anal., 220(3): 1283-1333, (2016)], the counterpart to the present work. To the best of our knowledge, this is the first work that rigorously establishes finite-time contact in a fluid-elastic structure interaction system, thereby validating the model to correctly capture near-contact dynamics.

The JMGT equation was put forward by Pedro Jordan~\cite{jordan2008nonlinear,jordan2014second}, also referring to earlier work by Moore and Gibson~\cite{moore1960propagation}, as well as Thompson~\cite{thompson} to amend the infinite speed of sound paradox of classical models of nonlinear acoustics such as the Westervelt and Kuznetsov's equation. Additionally to its physical significance (and of course related to it), it has given rise to a substantial body of mathematical literature – possibly even more than the above mentioned classical models. In this paper, we aim to provide a systematic (though inevitably incomplete) overview %and indicate some potential open questions. thereby focusing on well-posedness analysis of initial value and time periodic problems, memory and fractional attenuation as well as singular limits and – with one example each – control and inverse problems.

This note slightly strengthens the result of arXiv:2201.12295 on the linear instability of the Kerr Cauchy horizon. This strengthened result is used in the proof arXiv:2604.04877 of the non-linear instability of the Kerr Cauchy horizon.

In this paper, we establish global $C^{1, \alpha}$ regularity for viscosity solutions to a class of singular and degenerate fully nonlinear elliptic equations subject to oblique boundary conditions. Our work extends the findings in \cite{BKO25} to a broader class of equations, notably encompassing the singular case.

We consider the $SU(2)$ Self-Dual Yang Mills Higgs Lagrangian in 3 dimension. By adding a ”Gauge Mass” term to this YMH Lagrangian in the form of $L^2$ norm of the connection we break the gauge invariance and critical points are automatically fulfilling globally the Coulomb condition. We study the so called “large mass asymptotic”, which has the effect of ”squeezing” the monopoles. For any unit Higgs field data at the boundary we prove that minimizers of this Coulomb-Yang-Mills-Higgs Functional converge to harmonic maps into ${\mathbb S}^2$ extending this data. This asymptotic moreover is subject to concentration conpactness phenomena and the convergence is strong away from a 1 dimensional rectifiable closed concentration set. Then we prove that, having chosen a large enough coupling constant, the limiting minimal energy is converging towards the minimal Brezis-Coron-Lieb relaxed harmonic map energy for this boundary data. In the second part of the paper we examine a different asymptotic regime characterised by overloading monopoles. In this regime we prove that asymptotically, the magnetic field becomes exclusively longitudinal with a $U(1)$ abelian component along the Higgs Field while the Higgs field itself converges to a smooth absolute minimizer of a relaxation of the Faddeev-Skyrme functional of maps from ${\mathbb B}^3$ into ${\mathbb S}^2$. In the third part of the paper we study the behaviour of these configurations when the parameter in front of the Fadeev-Skyrme component respectively goes to zero and $+\infty$. In the first case one recovers the Brezis Coron Lieb relaxed energy at the limit while in the second case the minimal limiting energy is converging towards the minimal Dirichlet energy of maps into ${\mathbb S}^3$ whose projection by the Hopf fibration is equal to the fixed boundary data.

We present a formalization in Lean of the core interior De Giorgi–Nash–Moser theory for uniformly elliptic divergence-form equations with bounded measurable coefficients. The formalized results include local boundedness of weak subsolutions, the weak Harnack inequality for positive weak supersolutions, Moser's Harnack inequality for positive weak solutions, and interior Hölder regularity. This is, to our knowledge, the first machine-checked formalization of a major theorem in modern PDE theory. The development also required substantial new infrastructure for Sobolev spaces on bounded domains, weak solutions of elliptic equations, and quantitative regularity estimates. More broadly, it suggests that large-scale autoformalization of hard analysis in Lean is now within reach.

The three dimensional cubic defocusing nonlinear wave equation is known to be ill-posed for general low regularity initial data. Well-posedness can however be recovered globally in time on a probabilistic level, provided the Fourier coefficients of the random initial data follow a sub-Gaussian law, an assumption on which the available proofs rely heavily. In this article we perform numerical simulations of the one dimensional fractional cubic defocusing wave equation in a periodic setting, with low regularity random initial data drawn either from a Gaussian or from a heavy tailed Student law. Besides illustrating probabilistic well-posedness and norm inflation in both the energy subcritical and supercritical regimes, our simulations display no qualitative difference between the heavy tailed and the sub-Gaussian cases. This leads us to conjecture that probabilistic well-posedness holds under the sole assumption that the Fourier coefficients are centered with finite variance.

We are interested in the long-time behaviour of the kinetic Vicsek equation, rigorously derived as the mean-field limit~\cite{bolley2012meanfield} of a coupled system of~$N$ stochastic differential equations describing particles moving at unit velocity and aligning with their neighbours. We focus on the local-in-space version (that may for instance appear as a moderate interaction limit instead of mean-field), which is not a priori globally well-posed and could explode in finite time. Despite its simple expression, little is rigorously established about the behaviour of its solutions. We use hypocoercivity methods to show that finite time explosion does not occur in the vicinity of uniform and homogeneous equilibria in space below the critical threshold. We recast the now-classic~\cite{villani2009hypocoercivity} approach of modifying Sobolev-type norms by adding cross-terms, linked to commutators between the different operators appearing in the kinetic equation. However, the fact that the velocity space is the sphere adds significant subtleties and requires to develop an adapted algebraic framework of operators. Taking advantage of this new framework, we manage to perform an approach \textità la} Hérau~\cite{herau2007short} to show the nonlinear stability. Our main results are a quantitative decay estimate in the case of the whole space, despite the absence of control of the $L^1$ norm of the perturbation, and a gain in regularity at the nonlinear level which allows to have well-posedness and stability in the space~$H^{s,0}(\mathbb{R}^d\times\S)$ for some~$s<\frac{d}2$ (that is to say without a priori uniform bound in space on the~$L^2$ norm in velocity, that would come with Sobolev injection in the case~$s>\frac{d}2$).

We prove the exponential convergence to a unique invariant measure for locally damped nonlinear Schrödinger equations, perturbed by bounded noise acting on only two Fourier modes. To tackle the lack of smoothing effect, we introduce asymptotic compactness of linearized system to enhance the coupling method. Inspired by [14,33,39], we establish a new criterion for exponential mixing. Elements from global stability, nonlinear smoothing, and geometric control are combined when applying this criterion.

We study a two-dimensional incompressible vorticity equation on the torus driven by transport-type fractional Brownian noise with Hurst parameter $H \in (1/2,1)$. The model captures persistent, long-range correlated forcing consistent with inertial-range scaling laws and fractional Brownian approximations of turbulent fluctuations. A central ingredient of our approach is a version of the sewing lemma adapted to a class of integrands that includes, but is not limited to, transport-type structures. This result provides a flexible tool for constructing the Young integral and serves as a basis for analysing a wider class of stochastic partial differential equations. Using this approach, we establish existence and uniqueness of solutions via a fixed point argument and investigate statistical properties of the flow. In particular, we study quadratic functionals of the solution and derive an estimator for the Hurst parameter $H$.

In this note, we develop the first-order theory of optimal control problems with box constraints on the control. We emphasize the precise modification of Pontryagin's maximum principle when the admissible control set is compact, the projection/clamping formula for scalar quadratic Hamiltonians, the distinction between intrinsic projection inside the optimality system and post hoc truncation of an unconstrained solution, and the corresponding forward-backward sweep implementation. The presentation is pitched at senior PhD students who are already comfortable with variational arguments, adjoint systems, and basic nonlinear analysis. These notes are mainly based on the book “optimal control applied to biological models” of Suzanne Lenhart and John T. Workman.

We investigate singularly perturbed elliptic problems with multiplicative nonlocal diffusion terms subject to Robin boundary conditions. The diffusion depends on a global quantity of the solution, which introduces a nonlocal coupling between the global behavior of the solution and the boundary asymptotics. As the perturbation parameter tends to zero, we establish precise asymptotic expansions of the solutions that capture the structure of boundary layers coupled with the multiplicative nonlocal diffusion effect. Moreover, the interaction between the nonlocal diffusion and the boundary geometry manifests as refined higher-order terms wherein geometric quantities, such as the mean curvature, appear explicitly; our analysis thus quantifies the influence of global coupling on the boundary layer structure, extending classical singular perturbation theory to multiplicative nonlocal frameworks.

In this paper, we study the almost everywhere convergence of the cubic nonlinear Schrödinger flow to the initial data on $\mathbb S^2$, \begin{equation*} iu_t + \Delta_g u = |u|^2u, \quad (t,x)\in\R\times \S^2. \end{equation*} Inspired by the randomization method and the ansatz introduced by Burq, Camps, Sun, and Tzvetkov [Preprint, arXiv:2404.18229], we prove almost sure pointwise convergence almost everywhere for the nonlinear solution at very low regularity. This extends Compaan-Lucà-Staffilani [Int. Math. Res. Not. IMRN, (1) (2021), 596–647] to the spherical setting. We also provide a new necessary condition for the associated $L^p$ maximal estimate for the linear Schrödinger equation on $\S^2$. More precisely, we show that the $L^p$ maximal estimate fails for $s<\frac{1}{2}-\frac{1}{2p}$ with $p\ge 2$. In the special case $p=3$, our result matches the corresponding range in the $\R^2$ case, up to the endpoint, and improves the previous result of Chen-Duong-Lee-Yan [J. Math. Pures Appl. 163 (2022), 433–449].

This is the second of two papers devoted to the asymptotic behavior of solutions to the incompressible Navier-Stokes equations in a half-space with point vortex initial data. A major difficulty stems from the interaction between the point vortex initial data and the boundary, which complicates the derivation of a valid asymptotic expansion. To overcome this, we carry out a precise matching between the point vortex and boundary-layer profiles to accurately capture the correct viscous behavior of the vortex in the half-plane. Based on this matched asymptotic analysis, we decompose the vorticity into three components: vorticity near the point vortex, vorticity near the boundary, and vorticity in the transition layer. A key point is that each component must be analyzed in its own distinct region. On this basis, we establish refined estimates and thereby achieve the inviscid limit for the point vortex. Finally, we rigorously prove that solutions to the Navier-Stokes equations converge to the Lamb-Oseen vortex away from the boundary, while approaching the Prandtl boundary-layer system in the near-boundary region.

This is the first of two papers concerning the asymptotic behavior of the incompressible Navier-Stokes equations in a half-space at high Reynolds numbers, with initial data given by a point vortex. In the present work, we establish the existence and uniqueness of solutions subject to the non-slip boundary condition. This result was established in \cite{Ken} under the condition that the total mass is sufficiently small. Here, we eliminate the smallness assumption by analyzing the linearized operator near the point vortex and constructing a tailored functional framework-one designed to capture the distinct behaviors of the solution in the vicinity of the point vortex and the boundary, respectively.

We derive a unidirectional asymptotic model for one-dimensional blood flow in viscoelastic arteries. We prove local well-posedness of strong solutions in Sobolev spaces for general parameters and mean-zero periodic data. In the purely elastic BBM regime we further establish global existence and exponential decay for sufficiently small initial data. We also present a numerical study of the reduced model, including comparisons across different viscoelastic and amplitude regimes, and discuss the observed dynamics in connection with the continuation criterion.

We consider asymptotic behavior of solutions to the oblique-Dirichlet mixed boundary conditions without the strict monotonicity of the equation in the variable corresponding to the unknown function for "thin domains" i.e. when the N+1 dimensional domains collapse to an N dimensional domain. A global ellipticity condition in the limit equation is introduced.

In this article, we prove the existence of at least one positive solution for the mixed local-nonlocal semipositone problem \begin{equation*} \left\{ \begin{aligned} -\Delta_p u+ (-\Delta)^s_p u &= \lambda f(u) && \text{in } \Omega, u &= 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned} \right. \end{equation*} using mountain pass arguments, comparison principles and regularity principles.

This paper studies the asymptotic behavior of a one-dimensional Type II porous thermoelastic system with a conservative porous structure and local memory damping applied to the elastic component. Using frequency domain resolvent estimates, we prove polynomial decay of the associated semigroup. Our results clarify the effect of local memory damping and provide a unified framework for partially damped porous thermoelastic systems

In this paper, we investigate a system composed of two degenerate wave equations which are connected at one point. By introducing some inequalities on the weighted spaces and employing the frequency domain method, we prove that the system is polynomially stable,which depends on the degree of the degeneracy.