Papers

We construct an exhaustive family of constant spacetime mean curvature (STCMC) surfaces for initial data sets close to the anti-de Sitter-Schwarzschild hyperboloid. In particular, we obtain such a foliation as the long time limit of the volume preserving spacetime mean curvature flow starting from the constant mean curvature foliation constructed by Neves-Tian (Geom. Funct. Anal., 2009). As an application, inspired by the definition of STCMC center of mass for initial data sets proposed in the asymptotically Euclidean setting by Cederbaum-Sakovich (Calc. Var. PDE, 2021), we study the center of mass of an asymptotically hyperboloidal initial data set.

We consider an optimal insulation problem for a bounded domain in $\mathbb{R}^N$ driven by the $p$-Laplace operator ($p>1$). We model the convective heat transfer between the body and the environment, which corresponds, before insulation, to a nonlinear Robin boundary value problem. Assuming the body is surrounded by a thin layer of insulating material of size $\varepsilon^{\frac{1}{p-1}}$, we compute the $\Gamma$-limit of the governing energy functional as $\varepsilon \to 0^+$. Furthermore, we study the optimization of the heat content among all possible distributions of the insulating material with a fixed total mass. Finally, we highlight a concentration breaking phenomenon. Under a suitable non-degeneracy condition, if the boundary of the domain is connected or the external temperature profile is constant, the optimal insulating layer fails to cover the entire boundary whenever the total mass is sufficiently small. This is shown to be optimal: an explicit example provides that a disconnected boundary can trigger an anomalous double-phase transition, causing the insulation to fracture again even at intermediate mass regimes.

This study establishes Liouville-type theorems for indefinite quasilinear elliptic equations in the upper half-space. Additionally, we demonstrate the existence of solutions for this class of problems using the fibering method. Our approach relies on a novel weighted Sobolev embedding developed for the upper half-space.

In this paper, we study an elliptic problem involving a $p$-Laplacian operator and a potential well which is driven by a critical and singular nonlinearity. Under the limiting case of a parameter blowing up to $\infty$ yields solutions to a different problem where the effect of the potential well becomes negligible.

The spectral instability of traveling periodic water waves has been investigated for more than sixty years, since the seminal discovery of Benjamin and Feir. Despite an extensive literature, no rigorous theory has been available for arbitrary three-dimensional – longitudinal and transverse – perturbations. We establish the first rigorous description of the $3d $ unstable spectrum of small-amplitude gravity Stokes waves in deep water in a full neighborhood of the McLean resonant curves. Our results reveal that the Benjamin-Feir instability and the first longitudinal high-frequency isola originate from the same resonant interaction, hidden in the purely longitudinal setting. The dominant instabilities emerge for Fourier-Bloch parameters near the origin, corresponding to the $3d $ Benjamin-Feir modulational instability. Our approach provides quantitative bounds for the real parts of the unstable eigenvalues and establishes a computable necessary and sufficient criterion for the onset of instability near arbitrary high-frequency McLean curves. These results are enabled by three key innovations: ($i$) a Kato perturbative analysis allowing Lipschitz-type singularities of the linearized operator with respect to the Fourier-Bloch parameters; ($ii$) a polar-analytic KAM-type decoupling isolating the unstable eigenvalue pairs near the origin; and ($iii$) an analytic continuation argument in full neighborhoods of the McLean curves. A primary challenge is to establish fine regularity properties for the Dirichlet-Neumann operator conjugated via the Fourier-Bloch transform.

We study late-time identification of the Caputo order in a linear diffusion equation generated by a strictly positive self-adjoint operator with compact resolvent. For signed scalar observations \(M_\alpha(t)=\sum_n a_nE_{\alpha,1}(-\lambda_nt^\alpha)\) satisfying \(\sum_n|a_n|/\lambda_n<\infty\), we show that, after eigenspace grouping, every nontrivial observation has a finite first nonzero resolvent moment \(S_m=\sum_n a_n/\lambda_n^m\). A uniform differentiated large-argument expansion of the Mittag-Leffler factor yields eventual strict monotonicity of \(\alpha\mapsto M_\alpha(t)\) on admissible intervals avoiding the zeros of \(1/\Gamma(1-m\alpha)\), hence uniqueness from one sufficiently late scalar measurement. For two measurements, \(M_\alpha(\rho t)/M_\alpha(t)=\rho^{-m\alpha}(1+O(t^{-\alpha_0}))\), giving a log-ratio estimator with asymptotic-bias and relative-noise error bounds. For bounded observations, \(S_m=\langle\mathcal A^{-m}\varphi,h\rangle\); for a finite rod, the leading point-sensor condition is \((\mathcal A^{-1}\varphi)(x_*)\ne0\). Counterexamples show the sharpness of the exclusions and noise interpretation.

This article investigates the long-time behavior of (possibly infinite-dimensional) impedance passive systems under saturated output feedback and external disturbances. We assume that, in the absence of saturation and disturbances, the underlying linear output feedback exponentially stabilizes the system. Our main contribution is to show that fractional Sobolev regularity of the free output is preserved by the nonlinear feedback. More precisely, if the initial condition belongs to a suitable interpolation space associated with the linear closed-loop generator, then both the output and the state of the nonlinear closed-loop system inherit the corresponding fractional regularity. This regularity is sufficiently weak to be shared by the linear and nonlinear closed-loop systems, thereby avoiding the identification of the nonlinear generator domain. Combining this regularity result with observability estimates for the linear system yields a characterization of the asymptotic behavior of the nonlinear closed-loop system in the presence of disturbances. In particular, under the impedance passivity framework and exact observability and regularity assumptions, we establish a semi-uniform input-tostate stability property. The theory is illustrated by a multidimensional wave equation with nonlinear boundary damping.

We study the Cauchy problem associated with a class of triangular cross-diffusion systems of Shigesada-Kawasaki-Teramoto type. We develop a self-contained well-posedness theory in C 0 ([0, T ]; H s (T d )) based on regularity estimates for scalar Kolmogorov equations. The diffusion coefficient of each species depends only on species of lower index, yielding a hierarchical structure that allows for refined blow-up criteria. Finite-time singularities can occur only through the divergence of the L $\infty$ (T d ) norm of the solution. Assuming polynomial growth of the nonlinearities, this criterion is refined to an L p -based blow-up condition for some finite exponent p, yielding a substantially weaker obstruction to global existence than classical Sobolev blow-up criteria. The proof is achieved through refined tame estimates for composition in Sobolev spaces. As an application, we prove global existence of non-negative strong solutions for two-species systems with logistic-type reaction terms in dimensions d $\le$ 2.

We prove global existence for a degenerate Keller-Segel equation with small initial values in a critical Lorentz space.

We establish an analogue of the Brezis-Nirenberg local linking theorem for a class of Szulkin-type functionals arising from relativistic action principles. In this framework, compactness of Palais-Smale sequences is formulated with respect to a topology induced by the effective domain of the functional, replacing the classical strong Palais-Smale condition. The proof combines the original construction of the min-max geometry, based on a negative gradient flow, with the Ekeland-Lasry regularization. The main difficulty is that the regularized functional is naturally associated with the strong topology of the underlying functional space, whereas compactness for the original functional is formulated in the topology induced by the effective domain. We overcome this obstacle through a new perturbative construction that recovers the required min-max structure. We apply our abstract multiplicity result to two representative relativistic models: the Lorentz force equation, describing the dynamics of a charged particle in an electromagnetic field, and the Dirichlet problem for the prescribed mean curvature operator in Minkowski space. As a consequence, under natural assumptions, each problem admits at least two non-constant solutions.

We prove the finite-time stability of two-soliton solutions of the three-dimensional gravitational Hartree equation whose centers remain close to an elliptic solution of the Kepler two-body problem.

In this paper, we consider the global existence and properties of $H^1$ martingale solution to the Camassa-Holm equation with linear multiplicative noise under periodic boundary conditions. The solution is obtained as limit of regular viscous approximate solutions to parabolic SPDEs, which are constructed using the Galerkin approximations ans the stochastic compactness method. The proof of convergence to a solution argues via tightness of the laws of the viscous approximations and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. In particular, by means of the Girsanov-type transform for regular viscous approximations and the convergence of Skorokhod-Jakubowski representations, we are able to establish the one-sided supernorm estimate and space-time higher regularity of the first-order spatial derivative, and large-time behavior of the weak martingale solution in the stochastic framework.

We prove the existence of weak solutions to the surface Beris-Edwards model for nematic liquid crystals posed on a $d$-dimensional ($d \in \{2,3\}$) closed hypersurface of class $C^{2,1}$. This thermodynamically consistent model, recently introduced by Bouck, Nochetto and Yushutin (2024), couples the incompressible tangent Navier-Stokes equations with a kinematic equation for the Q-tensor field that encodes the orientation of the liquid crystal particles with a general state of orientational order. Extending ideas by Abels, Dolzmann and Liu (2014) and Guillén-González and Rodríguez-Bellido (2015) for the Beris-Edwards model in flat domains, we design a Faedo-Galerkin scheme based upon eigenfunctions of an appropriate tangent Stokes operator and tensor-valued Laplace-Beltrami operator and recover a weak solution via standard compactness arguments.

In this paper, we prove that the solution map of Camassa-Holm equation with linear multiplicative noise $$ \left\{ \begin{array}{l} {\rm d}u+(u\partial_xu+\partial_xP[u])\,{\rm d}t=\beta u\,{\rm d}W, u(0,x)=u_0(x), P[u]=(1-\partial_x^2)^{-1}\left(u^2+\frac 1 2(\partial_x u)^2\right) \end{array} \right. $$ depends almost surely continuously on the deterministic initial data in $H^s$ for $s>3/2$. Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise.

We consider the conditional regularity of the mild solution $v$ of the $3-D$ incompressible Navier-Stokes equations with initial data $v_0\in \dot{H}^{\frac 1 2}$ and vorticity $\Omega_0\in L^{r_0}$ for some $r_0\in (1,2)$. We prove that if the solution associated with initial data $v_0$ blows up at a finite time $T^\ast$, then for any $2<p<\infty$, and any unit vectors $e$ in $\mathbb{R}^3$, the integral $$\int_0^{T^\ast}\left\Vert (v(t)|e)_{\mathbb{R}^3}\right\Vert_{\dot{B}^{\frac 1 2+\frac 2 p}_{2,\infty}}^p{\rm d}t$$ blows up at $T^\ast$. The conclusion improves the recent results in Chemin et al. (Arch Ration Mech Anal 224(3):871-905, 2017) and Han et al. (Arch. Rational Mech. Anal. 231:939-970, 2019).

We consider the one-dimensional cubic nonlinear Schrödinger equation with a non-generic real-valued external potential $V$. We prove almost global-in-time quantitative bounds for small solutions. More precisely, small initial data of size $\varepsilon$ in a weighted Sobolev space give rise to solutions with the sharp decay rate $t^{-1/2}$ in $L^{\infty}_x$ up to time $\exp(\frac{1}{c\varepsilon^{2}})$. The main novelty of our result is that no additional symmetry assumption is imposed on $V$. First, we use a modification of the standard distorted Fourier transform basis to resolve the possible discontinuity at zero energy due to the presence of a resonance. Then, following the work of Chen and Pusateri, we use smoothing estimates in the setting of non-generic potentials to analyze the low frequency structure of the (modified) nonlinear spectral distribution. A key novel ingredient is a Fourier restriction type inequality that handles low frequency contributions not amenable to the approach of Chen and Pusateri, and which is central to establishing the quantitative bounds.

We study a small-denominator mechanism in weakly nonlinear dispersive dynamics. After Fourier decomposition, a nonlinear dispersive equation becomes an infinite system of weakly coupled oscillators. Higher-order correction terms may then contain infinite families of nonresonant Fourier interactions whose detunings tend to zero. Such families do not produce exact secular terms, but their accumulated contribution may grow as a power of time. We call this effect subresonant accumulation. The rigorous part of the paper is the analysis of a model forced oscillator and of an abstract subresonant Duhamel sum. If the detuning and coefficients have the form $\Delta_n\sim c n^{-p}$ and $B_n\sim b n^{-\kappa}$, then the accumulated contribution grows as $t^{1-\alpha}$, where $\alpha=(\kappa-1)/p$. We then show how this mechanism appears in a quartic Fourier family for the Klein–Gordon dispersion law. For the full nonlinear partial differential equation we formulate a conditional approximation result: provided that all remaining resonant and almost resonant interactions are controlled, the subresonant term gives the leading long-time correction.

We prove that a complete Berwaldian manifold $\left(M,F\right)$ admitting a nontrivial conformal circle preserving transformation (\cpt for short) must be Riemannian, provided that it has a dense subset on which no flag curvature vanishes (in particular, if $(M,F)$ has positive or negative flag curvature).

We investigate a coupled system of partial differential equations modeling the interaction between Brinkman flow, biofilm evolution, and nutrient transport in a porous medium. The model captures the mutual influence between the fluid velocity and the biofilm through drag and diffusion coefficients that depend on the local biofilm volume fraction. A hard constraint on the admissible range of the biofilm fraction is incorporated through the subdifferential of an indicator functional, which leads naturally to an evolution variational inequality formulation for the biofilm dynamics. Assuming standard coercivity, ellipticity, and growth conditions on the model coefficients and reaction terms, we prove the global-in-time existence of weak solutions. The analysis relies on a decomposition of the system into three interconnected subproblems: the Brinkman equation with a fixed biofilm profile, the constrained biofilm evolution treated through maximal monotone operator theory, and the nutrient equation viewed as a semilinear parabolic problem. These components are then coupled through a Leray–Schauder type fixed-point argument, with the passage to the limit justified by Aubin–Lions and Simon compactness results. We further establish the nonnegativity of the nutrient concentration under a natural quasi-positivity assumption on the reaction term. Finally, we provide conditional uniqueness results for weak solutions in two spatial dimensions under additional smallness assumptions.

In this paper, we study blowup solutions of an important class of Chern-Simons systems. We first show that when blowup of mean-field type occurs, the corresponding blowup solution is unique under natural geometric assumptions. We also establish the non-degeneracy of the linearized system around these blowup solutions. To prove these main results, we carry out a precise blowup analysis, so that the asymptotic description of the solutions reveals the curvature information needed for the uniqueness and non-degeneracy results. Compared with related work on similar problems, our estimates are more delicate and technically involved.

We prove the Ashbaugh–Benguria reciprocal-gap conjecture for the Dirichlet Laplacian in every dimension $N\ge2$. Specifically, if $\Omega\subset\mathbb R^N$ is a bounded domain and $$ 0<\lambda_1(\Omega)<\lambda_2(\Omega)\le\lambda_3(\Omega)\le\cdots $$ are its Dirichlet eigenvalues, then $$ \sum_{i=1}^{N} \frac{\lambda_1(\Omega)} {\lambda_{i+1}(\Omega)-\lambda_1(\Omega)} \ge \frac{N}{j_{N/2,1}^2/j_{N/2-1,1}^2-1}, $$ where $j_{\mu,1}$ denotes the first positive zero of the Bessel function $J_\mu$ of the first kind of order $\mu$. We also characterize the equality case: equality holds precisely when $\Omega$ agrees with a Euclidean ball up to a set of Sobolev $H^1$-capacity zero. In particular, among bounded Lipschitz domains, equality holds if and only if $\Omega$ is a Euclidean ball.

We consider the Hasegawa-Mima equation in its “Euler-like” velocity form: \[\partial_t(u-\Delta^{-1}u)+(u\cdot\nabla)u-u^\perp\log n_0=0,\] $n_0$ being the time-independent function appearing in the particle count $n=n_0e^{\frac{e\varphi}{T}}$, and $u$ being the drift velocity $\nabla^\perp\varphi=-\nabla\varphi\times\hat z$. Adapting the notion from Lions' book on the Euler equations, we prove the existence of dissipative solutions for this equation for any $L^2$ divergence free initial condition $w\in L^2(D)$, for $D=\mathbb T^2$ and $D\subset\mathbb R^2$ a bounded $\mathcal{C}^1$ domain.

We use the Laplacian in n-dimensional spherical coordinates (n>1) to write the divergence of a vector field defined on radially symmetric domains in the context of vector calculus. We apply straightforward equations of vector calculus with the nabla operator and the transformation matrices from Cartesian to spherical polar coordinates. One needs the divergence of a vector field e.g. to prove that vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls. Our divergence formula in partial derivatives in n-dimensional spherical polar coordinates is an important step in a future verification of further Stokes eigenfunctions on those domains.

We investigate the influence of routing strategies and speed limit policies on optimal solutions in traffic emission models. Building on a first-order macroscopic traffic model coupled with an advection-diffusion model, we formulate single- and multi-objective optimization problems to simultaneously maximize traffic efficiency and minimize air pollution. We compare three control scenarios: optimizing only the routing strategy, optimizing only the speed limit policy, and optimizing both simultaneously. Numerical experiments on a small road network demonstrate that speed limit policies consistently achieve larger reductions in emissions and greater gains in traffic efficiency than routing strategies. Multi-objective optimization reveals the trade-off between the two goals and confirms that including speed limits in the control set yields Pareto-optimal solutions that are strictly superior to those obtained by routing control only. Our results provide quantitative guidance for traffic management seeking to balance mobility and environmental objectives.

We study the three-dimensional incompressible free-boundary ideal magnetohydrodynamic (MHD) equations with surface tension and a closed free surface. Our first result establishes $H^3$ a priori estimates in general bounded domains, without graph structure, periodicity, or simple connectedness; in particular, for surface-tension ideal MHD in general domains this lowers the previously available threshold from $H^6$. Compared with the free-boundary problem for incompressible Euler equations, the feature is that the Lorentz force enters the elliptic pressure estimates, and the frozen-in magnetic field must preserve the tangential boundary constraint. Using these estimates, we prove a refined finite-time blow-up criterion for $H^3$ solutions that separates topological self-intersection, loss of boundary regularity, blow-up of the normal velocity, and interior MHD blow-up. The interior condition has an intrinsic magnetic-field asymmetry: besides $\|\nabla u\|_{L^\infty}$ and $\|\nabla h\|_{L^\infty}$, with $u$ and $h$ denoting the velocity and magnetic field, respectively, it requires the additional control of $\|\nabla^2 h\|_{L^2}$, a quantity arising from the Lorentz-force contribution to the pressure estimates and having no velocity analogue. Finally, we construct regular initial data whose solutions develop finite-time boundary self-intersection while the Sobolev regularity and curvature remain controlled up to the contact time. Thus, neither surface tension nor the ideal magnetic coupling precludes topological self-intersection of the free boundary.

W}e study the equation $P(i\nabla)u=0$ on $\mathbb{R}^d$ for a class of admissible symbols $P$ whose zero set is the unit sphere $S^{d-1}$ and which vanish there to some finite order. Working in the framework of Lizorkin distributions, and hence without any boundedness or decay hypothesis on $u$, we give a complete classification of the solutions: $u$ solves $P(i\nabla)u=0$ if and only if $\hat{u}$ is a multi-layer distribution on $S^{d-1}$ of order at most $N$. Alternatively, $u$ solves $P(i\nabla)u=0$ if and only if $(1+\Delta)^{N+1}u=0$ if $P$ satisfies a flatness condition. The proof recasts the equation as a division problem and combines the order of vanishing of $P$ with the structure theorem for distributions. This unifies and extends known Helmholtz-type rigidity results, which correspond to a simple zero on the sphere, to symbols with zeros of arbitrary finite order.

We study a notion of fractional $s$-mass for codimension-two currents on closed Riemannian manifolds, defined via energy minimization with a prescribed Jacobian constraint. We prove equi-coercivity and $\Gamma$-convergence, with respect to the flat topology, of the $s$-mass on general codimension-two currents. We also prove several additional results for fixed $s$. We establish improved regularity for $s$-harmonic maps that are minimizing among competitors with vanishing Jacobian and show that their singular set has Minkowski dimension at most $n-3$. Moreover, we show that the $s$-mass defined via weak linking, as recently introduced by the authors, agrees with the prescribed Jacobian formulation used here, clarifying the extent to which the $s$-mass depends, or ultimately does not depend, on the way singularities are prescribed.

This paper is devoted to the shape analysis of the energy of a caloric family of Schauder functions defined on a bounded perforated domain $\Omega^o \setminus \overline{\Omega^i[\phi]}$ of $\mathbb{R}^n$, where the outer boundary is fixed, and the inner boundary is obtained by a $C^{1,\alpha}$-perturbation $\phi$ of the boundary of a reference cavity $\Omega^i$. Without imposing any boundary conditions, we prove that in a suitable neighborhood of the identity $\phi_0$, the domain-to-energy map is of class $C^{\infty}$. The proof is based on the construction of a global diffeomorphism, smoothly depending on $\phi$, from the reference annulus onto the perturbed one and on suitable regularity and smoothness assumptions on the pull-back family onto the reference domain. We then apply our main result to two boundary value problems: a nonlinear mixed Robin-type problem and a linear Dirichlet problem. After recalling some known existence and shape analysis results for the solutions, we prove that the corresponding domain-to-energy map is of class $C^{\infty}$. The proof is based on a decomposition of the fixed domain into near, intermediate, and far regions relative to the cavity, and on the smooth dependence of the layer heat potentials upon support perturbations.

A Minkowski normed space is the Euclidean space equipped with a (possibly asymmetric) uniformly convex and smooth norm, forming a particular class of Finsler manifolds. We construct a stochastic process with one-dimensional time marginal densities given by the fundamental solution to the nonlinear Finsler heat equation in Minkowski normed spaces. This process is constructed as a solution to a singular McKean–Vlasov stochastic differential equation and constitutes a nonlinear Markov process in the sense of McKean. Furthermore, we show that solutions to this stochastic differential equation are pathwise unique, and thus probabilistically strong solutions, though the equation has singular coefficients beyond the subcritical regime. Since our construction is a natural extension of the construction of standard Brownian motion from the standard heat kernel, we call this process Brownian motion in Minkowski normed spaces. To the best of our knowledge, this is the first construction of stochastic processes associated with nonlinear heat equation in Finslerian spaces.

The filtering method applied at the level of the Euler-Maxwell-Two-Fluid system produces a Fast Limit Model (FLM) which captures up to the electron depth essential features of plasma dynamics. In the case of prepared data, the discussion reduces to the eXtended MagnetoHydroDynamic (XMHD) framework of physicists, which involves the density __, the velocity u and the magnetic field B as state variables. By contrast, for unprepared data, an electric field E is created by resonances, and it participates to the time evolution. It turns out that FLM is a well-posed system on (__, u, E, B), extending XMHD, and implying a mechanism of interactions between (__, u, B) and E which can convert a part of the energy carried by (__, u, B) into electric energy.