Papers

We consider a branched transport type problem with weakly imposed boundary conditions, which can be seen as a blown-up version of a reduced model for type-I superconductors in the regime of vanishing external magnetic field. We prove that if the irrigated measure is (locally) Ahlfors regular then it is of dimension at most $8/5$ in agreement with the conjecture by Conti, the third author and Serfaty.

In this paper, we focus on one-dimensional vertical infiltration, assuming constant diffusivity and a quadratic relationship between hydraulic conductivity and water content. Under these assumptions, Richards' equation reduces to Burgers' equation, which we then linearize via the Hopf-Cole transformation. This turns the initial boundary value problem into a diffusion equation on a finite interval with mixed boundary conditions. To solve it, we use the Unified Transform Method (also known as the Fokas method). This approach gives an explicit integral representation of the solution, and when evaluated numerically, the results match classical Fourier series solutions exactly, but with better convergence and stability. Two examples from hydrological applications are examined.

In this work, we study a phase transition model in atmospheric dynamics, inspired by the works [6,14,15], which analyze the primitive equations governing the evolution of velocity, temperature, and specific humidity. The main difficulty arises from the presence of a multivalued discontinuous nonlinear term in the temperature and in the humidity equations, describing the formation of precipitations, which becomes active under supersaturation conditions. To overcome this issue, we introduce a regularized formulation that ensures the existence and uniqueness of approximate solutions. By employing classical compactness arguments, we then establish the existence of a strong solution to the original model. Additionally, we establish uniqueness under a conditional and physically meaningful assumption. This approach allows us to provide a rigorous justification of the tropical climate model on the whole space $\mathbb{R}^2$, while avoiding the introduction of a viscosity term in the humidity equation.

We consider the defocusing nonlinear Schrödinger equation in the energy-subcritical case, and investigate the dependence of the solution upon the power of the nonlinearity. Special attention is paid to the global in time description. The main three aspects addressed, in the decreasing order of difficulty, are the limit when the total power tends to one, along with the connection with the logarithmic Schrödinger equation, the description when long range effects may be present, and the continuity of the scattering operator in the short range case. This text resumes the presentation given by the first author at École polytechnique for the Laurent Schwartz seminar, in May 2026.

This paper investigates the Newton's problem of minimal resistance for a body moving through a fluid whose density decreases exponentially with altitude. We prove the local existence and regularity of radial solutions $u(r)$ satisfying the initial conditions $u(0)=u'(0)=0$ using a fixed-point theorem. We show that the maximal domain of the solution is finite, $[0, r_M)$, terminating at a critical slope $u'(r_M) = \frac{1}{\sqrt{3}}$.

We develop a deterministic large-time mechanism yielding Cesàro asymptotic observability inequalities from moving localized observations for conservative evolutions. On each observation interval, exact convexification on a compact measured homogeneous space replaces full observation on the whole observation manifold by a finite convex combination of translates of one prototype subset. A switching realization theorem then turns that static design into a genuinely moving observer, while a Hilbertian tail-reduction proposition shows that interval estimates proved only on growing spectral windows still recover the full conserved energy after Cesàro averaging. The resulting design-to-observability chain applies to interior observations for wave, Klein-Gordon, and Schrödinger equations on compact measured homogeneous manifolds, to moving boundary caps on the Euclidean ball, and to a singular almost-separated gas-giant boundary model. The framework is especially relevant when each instantaneous observation set is too small for one to expect a finite-time GCC or time-dependent GCC statement.

We establish an averaging principle for a structural multiscale stochastic nonlinear fractional Schrödinger system on the one-dimensional torus driven by a multiplicative Wiener noise. The slow component is governed by a fractional Schrödinger operator with a general polynomial nonlinearity, while the fast component evolves on a shorter time scale and exhibits dissipative diffusion, nonlinear interactions, and stochastic forcing. Under suitable dissipative assumptions, we have shown that, as the scale separation parameter tends to zero, the slow component converges strongly to an effective stochastic fractional Schrödinger equation. The effective drift is obtained by averaging the coupling term with respect to the unique invariant measure of the frozen fast dynamics. The proof relies on uniform a priori estimates, ergodicity of the fast equation, Hölder time regularity of the slow component obtained via a vanishing viscosity method, and a Khasminskii-type time discretization argument adapted to fractional dispersive operators. The analysis is technically challenging due to limited smoothing of the fractional Schrödinger semigroup and the presence of general polynomial nonlinearities, which are handled through refined estimates and viscosity approximation.

We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the conditions \[ \lambda_1(-\gamma\Delta+\mathrm{Ric})\geq 0,\qquad H_{\partial M}\geq 0, \] and in the sharp range $0\leq \gamma<4$ if $n=2$, and $0\leq\gamma<\frac{n-1}{n-2}$ if $n\geq3$, a (possibly noncompact) complete manifold with disconnected boundary, with at least one compact boundary component, must split isometrically as a product $[0,L]\times \Sigma$. Our second main contribution is a topological rigidity result for the relative fundamental group $\pi_1(M,\partial M)$, combined with a deep theorem of Lawson–Michelsohn. We prove that, in dimensions $n\neq4$, any compact manifold with boundary satisfying the two inequalities above, with at least one of them strict, admits a metric with positive sectional curvature and strictly mean-convex boundary, provided $\gamma\geq0$ if $n=2$, and $0\leq\gamma\leq\frac{n-1}{n-2}$ if $n\geq3$. This range of $\gamma$ is sharp for the latter result to hold.

In this paper, around a global smooth irrotational solution to the classical isentropic compressible Euler-Poisson system, we construct classical solutions to the one-species relativistic Vlasov-Maxwell-Boltzmann system on any finite time interval $[0,T]$, and rigorously justify the combined hydrodynamic and Newtonian limits to the Euler-Poisson system. In particular, this yields a rigorous derivation of the compressible Euler-Poisson system, whose Poisson coupling induces an instantaneous electrostatic response and thus no longer preserves a strict finite-speed propagation structure, from a relativistic kinetic model with finite propagation speed. The analysis is based on a Hilbert expansion in $\varepsilon$ for the relativistic Vlasov-Maxwell-Boltzmann system, an asymptotic expansion in $\mathfrak{c}^{-1}$ for the relativistic Euler-Maxwell system, and estimates for both the expansion coefficients and the remainder terms that are uniform in $\mathfrak{c}$ and $\varepsilon$, without imposing any a priori relation between these two parameters.

We study the two-dimensional Hele-Shaw problem with point injection for star-shaped domains. We reduce the system to a nonlocal parabolic equation of the interface, and for arbitrary Lipschitz initial interface away from the source, we prove global well-posedness of the interface equation in a strong sense. We also introduce a viscosity-solution framework for the interface equation and relate it to the classical viscosity theory for the Hele-Shaw problem.

Medvedev proved that the critical spherical catenoid $Sigma_a$, the rotationally symmetric free boundary minimal annulus in a geodesic ball $B^3(r(a))$ in $H^3$ from the family of Mori and do Carmo-Dajczer, has Morse index at least 4, and conjectured equality. For each $a>1/2$ we establish three analytic results about $Sigma_a$. (I) Robin nullity and index in mode $|k|=1$. The Robin nullity of the Jacobi operator $L_{\Sigma_a}=\Delta_g+(|II|^2-2)$ in angular Fourier mode $|k|=1$ equals $2$, with kernel spanned by the Killing–Jacobi fields associated to the rotations $L_{12},L_{13}\in\mathfrak{so}(3,1)$ that fix the geodesic axis of $\Sigma_a$ and send $\partial B^3(r(a))$ to itself. The radial profile admits the closed form $f_*(s)=\partial_s\Phi_a^0(s,0)=\frac{d}{ds}[A(s)\cosh\varphi(s)]=\sinh r(s)\cdot r'(s)$, where $r(s)$ is the geodesic distance from $p_0=(1,0,0,0)$. By Sturm–Liouville theory, the Robin Morse index of $\Sigma_a$ in mode $|k|=1$ also equals $2$, refining the lower bound of Medvedev. (II) Asymptotic radius. The boundary radius satisfies $r(a)=\tfrac{3}{2}\log a+d_\infty+o(1)$ as $a\to\infty$, with $d_\infty=\log[\sqrt{2}\,\Gamma(1/4)^2/\pi^{3/2}]=\log[2\sqrt{2\pi}/\Gamma(3/4)^2]$. The closed form for $d_\infty$ follows from a Beta-function evaluation of $I_\infty=\int_0^{\infty}\cosh(2t)^{-3/2}\,dt$. (III) Degenerate limit. As $a\to(1/2)^+$, $r(a)=c_*\sqrt{a-1/2}\,(1+o(1))$ with $c_*=\sigma_*\cosh\sigma_*$, where $\sigma_*$ is the unique positive fixed point of $\sigma=\coth\sigma$. The proof of (I) follows the mode-by-mode strategy of Devyver for the Euclidean critical catenoid, with $\mathfrak{so}(3,1)$ replacing $\mathfrak{so}(3)$, supplemented by the closed-form identification $f_*=\partial_s\Phi^0$ specific to the hyperbolic ambient. The proof of (II) is a Laplace-type asymptotic analysis of the implicit free boundary condition.

We present a short and flexible improvement-of-flatness argument adapted to the setting of exterior domains, where one is naturally led to work with annuli instead of balls. As a model application in the classical setting of minimal surfaces, we give an alternative proof of the end-structure and asymptotics for finite Morse index minimal hypersurfaces with Euclidean area growth in low dimensions. The method is largely PDE-based and general in its application. Suitable variants have been employed in Bernoulli and Allen–Cahn settings.

Transformers with self-attention modules as their core components have become an integral architecture in modern large language and foundation models. In this paper, we study the evolution of tokens in deep encoder-only transformers at inference time which is described in the large-token limit by a mean-field continuity equation. Leveraging ideas from the convergence analysis of interacting multi-particle systems, with particles corresponding to tokens, we prove that the token distribution rapidly concentrates onto the push-forward of the initial distribution under a projection map induced by the key, query, and value matrices, and remains metastable for moderate times. Specifically, we show that the Wasserstein distance of the two distributions scales like $\sqrt{{\log(\beta+1)}/{\beta}}\exp(Ct)+\exp(-ct)$ in terms of the temperature parameter $\beta^{-1}\to 0$ and inference time $t\geq 0$. For the proof, we establish Lyapunov-type estimates for the zero-temperature equation, identify its limit as $t\to\infty$, and employ a stability estimate in Wasserstein space together with a quantitative Laplace principle to couple the two equations. Our result implies that for time scales of order $\log\beta$ the token distribution concentrates at the identified limiting distribution. Numerical experiments confirm this and, beyond that, complement our theory by showing that for finite $\beta$ and large $t$ the dynamics enter a different terminal phase, dominated by the spectrum of the value matrix.

We study the generalized Stokes operator \begin{equation*} \bsXi \ede \bsXi _{V,V_0} \ede \left(\begin{array}{ccc} \bsL + V & \nabla \\ \nabla^* & -V_0 \end{array}\right) \end{equation*} on a domain with straight cylindrical ends $\Omega$ using the method of layer potentials on $M \supset \Omega$. The operator $\bsXi_{0, 0}$ is the classical Stokes operator. Under suitable positivity assumptions on $V$ and $V_{0}$, we prove that $\bsXi$ is Fredholm. This allows us then to define the single- and double-layer potentials $\bsS$ and $\frac12 + \bsK$. Under further positivity assumptions, we prove that $\bsS$ and $\frac12 + \bsK$ are also Fredholm. Under slightly stronger assumptions on $V$ and $V_{0}$, we prove the invertibility of the operators $\bsXi$, $\bsS$, and $\frac12 + \bsK$. The invertibility of these operators leads to well-posedness results for the associated (linear) Stokes boundary value problem with Dirichlet boundary conditions on $\Omega$. The proofs of these results required us to develop many related tools. In particular, we develop an “algebra tool kit” to deal with limit and jump relations of layer potentials. We also develop Green formulas and energy estimates for our generalized Stokes operator $\bsXi$ on manifolds with straight cylindrical ends, which requires a careful geometric study of the related differential operators, such as the deformation operator $\Def$. For completeness, we review suitable classes of pseudodifferential operators on manifolds with straight cylindrical ends that were studied in some previous papers of ours (including “The Stokes operator on manifolds with cylindrical ends,” J. Diff. Equations, 2024). As an application, we prove the well-posedness result for the Dirichlet problem for the generalized Navier-Stokes system with small data on a domain with cylindrical ends.

We prove a sharp convergence theorem for the Yang-Mills flow on an $\mathrm{S}\mathrm{U}(r)$-bundle over a locally hyperKähler ALE 4-manifold. Our main result is a noncompact version of the "parabolic gap theorem" previously established by the authors.

We establish a new class of entropy structures for \(3\)-wave kinetic equations with a broad family of interaction weights. Unlike the classical entropies arising from detailed balance, these estimates are generated by a one-sided algebraic balance condition encoded in the interaction weights. To the best of our knowledge, this family of entropy estimates has not previously appeared in the physical literature on wave turbulence. These estimates form the central a priori mechanism of the paper and are the key ingredient in the construction of global weak \(L^1_{\mathrm{loc}}\) solutions. We also prove a long-time rigidity result, showing that the solutions obtained by this entropy compactness method relax locally to the zero equilibrium as \(t\to\infty\).

We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading in particular to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.

We establish a Brézis-Oswald-type spectral principle for semilinear equations \[ -Lu=f(x,u), \] where $L$ is the generator of a positive $C_0$-semigroup on $L^p$ with compact positivity-improving resolvent. Neither symmetry, variational structure, nor regularizing properties such as ultracontractivity or smoothing are assumed. Let $a_0$ and $a_\infty$ denote the asymptotic slopes of the nonlinearity at zero and at infinity, and let $\lambda_1(a)$ denote the generalized principal eigenvalue associated with the perturbed operator $-L-a$. We prove that \[ \lambda_1(a_0)<0<\lambda_1(a_\infty) \] implies that the equation admits a strictly positive solution. The proof develops a potential-theoretic sub- and supersolution framework based on the order induced by supermedian functions and combines a Deny-type compactness theorem, a Kato-type inequality, and a Doob transform. The construction also yields an order-preserving selection of solutions and allows the lower control on the nonlinearity to be relaxed. If $y\mapsto f(x,y)/y$ is strictly decreasing and $a_0$ is bounded, the spectral condition is necessary as well, and the strictly positive solution is unique.

This paper resolves the global regularity problem for the three-dimensional compressible magnetohydrodynamics (MHD) equations in the three-dimensional whole space, in the presence of a background magnetic field. Motivated by geophysical applications, we consider an anisotropic compressible MHD system with weak dissipation in the $x_2$ and $x_3$ directions and small vertical magnetic diffusion. By exploiting the stabilizing effect induced by the background magnetic field and constructing a hierarchy of four energy functionals, we establish global-in-time uniform bounds that are independent of the viscosity in the $x_2$ and $x_3$ directions and the vertical resistivity. A key innovation in our analysis is the development of a two-tier energy method, which couples the boundedness of vertical derivatives with the decay of horizontal derivatives. The analysis of time scale, together with global regularity estimates and sharp decay rates, enable us to rigorously justify the vanishing dissipation limit and derive explicit long-time convergence rates to the compressible MHD system with vanishing dissipation in the $x_2$ and $x_3$ directions and no vertical magnetic diffusion. In the absence of magnetic field and background magnetic field, the global-in-time well-posedness and vanishing viscosity limit for the 3D compressible Navier-Stokes equations with only one direction dissipation remains a challenging open problem. This work reveals the mechanism by which the magnetic field enhances dissipation and stabilizes the fluid dynamics in the global well-posedness and vanishing viscosity limit.

For $\varepsilon>0$, let $\phi^\varepsilon$ be the solution of the ergodic problem \[ \frac12 |D\phi^\varepsilon|^2+F(x)-\varepsilon\Delta\phi^\varepsilon=c(\varepsilon) \qquad \text{on } \mathbb{T}^n, \] normalized by $\phi^\varepsilon(0)=0$. We construct a one-dimensional example with $F\in C^3$ for which the vanishing-viscosity limit $\lim_{\varepsilon\to0}\phi^\varepsilon$ does not exist. This gives a negative answer to a problem proposed by Jauslin, Kreiss, and Moser [10].

We study the equation of one-dimensional quasistatic nonlinear viscoelasticity with Dirichlet boundary conditions, in the particular case that the underlying dissipation geometry (provided by the viscosity) is comparable to the Bhattacharya metric on probability densities. We establish a global existence result for weak solutions, with an approach based on a spatial discretization allowing us to work directly with the Riemannian metric associated to the viscosity. Strong convergence of spatially discrete solutions is shown directly - this is possible thanks to Lipschitz estimates achieved locally on energy sublevels enabled by an explicit derivation of the stretching of tangent vectors under the flow in the discrete setting and the relationship to the Bhattacharya metric. We furthermore prove gradient-flow representations for the solutions: they are curves of maximal slope and, under a global convexity hypothesis on the energy sublevels, we prove they satisfy a metric evolutionary variational inequality.

We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality \[ -\Delta_p u\ge \sigma(x)u^q \] on infinite locally finite connected weighted graphs, where $1<p<\infty$ and $q>p-1$, $\sigma$ is a nonnegative Radon measure. Under the non-$p$-parabolic setting, we show that every nonnegative solution is identically zero, provided the volume of intrinsic balls satisfy \[ \int_1^\infty \frac{r^{\frac{pq}{p-1}-1}} {\nu(B_\rho(o,r))^{\frac{q-p+1}{p-1}}} \dd r =\infty, \] This criterion recovers the known sharp pointwise critical volume-growth threshold and is strictly more flexible, since it allows irregular growth and does not require uniform upper bounds at every large radius. The proof adapts the finite-network current method to the $p$-Laplace setting, combining a path decomposition with one-dimensional Hardy estimates, $p$-parallel-sum bounds across metric cuts, and the global $p$-Green function furnished by non-$p$-parabolicity.

This article addresses the solvability of the three-dimensional divergence-curl problem with a no-slip boundary condition. A solvability criterion is derived as an orthogonality condition of the vorticity function to pseudo-harmonic fields. A countable family of such fields, sufficient for the solvability of the three-dimensional problem in the exterior of a sphere, is also presented.

In this paper, we prove global $ C^{1}$ regularity for Dirichlet problem of a class of degenerate fully nonlinear elliptic equations on $ C^{2}$ domain. This corresponds to the boundary counterpart of the interior $ C^{1}$ regularity results by \cite{APPT22} and \cite{AN25}. By an example, we show that $ C^{1, \alpha}$ regularity of boundary datum is sharp within the scale of Hölder spaces. Furthermore, we also provide boundary $ C^{1, \beta}$ regularity for a class of singular fully nonlinear elliptic equations.

The problem of continuous data assimilation for semilinear parabolic equations based on partial observations corrupted by noise is investigated. The noise is allowed to be multiplicative, with additive noise arising as a special case. In a general Gelfand triple framework, an abstract theory for the nudging equation is developed that covers both weak and strong formulations. Mean square convergence of the assimilation error is proved under suitable assumptions, and, under additional integrability conditions on the noise, a uniform almost sure convergence result is established. Finally, the framework is applied to several PDE models, including the 2D Navier-Stokes, 2D magnetohydrodynamics, 2D quasi-geostrophic, and 1D Allen-Cahn equations.

We investigate the global well-posedness and asymptotic behavior of $L^2$-solutions to stochastic nonlinear Schrödinger equations with multiplicative noise driven by continuous square integrable martingales with density. Our approach relies on a rescaling transformation that converts the stochastic system into a random nonlinear Schrödinger equation with a potential acting as a damping term. Unlike the standard Brownian motion case, this induced potential plays a critical role in the dynamics. We establish the global existence of solutions and prove the pathwise exponential decay of the $L^2$-norm. Crucially, the strict positivity of the decay rate is intrinsically induced by the density of the martingale\rq{}s quadratic variation. This result generalizes the stabilization known for standard Brownian motion, thereby characterizing the stabilizing effect of the martingale noise.

Subject of consideration is the modelling and analysis of a capillary-driven three-dimensional rimming-flow problem. We present the derivation of a fourth-order quasilinear degenerate-parabolic partial differential equation for the height $h > 0$ of a fluid film coating the inner wall of a cylinder that rotates around a horizontal axis. The equation arises from a rescaled Navier-Stokes system for thin fluid films by means of a lubrication approximation and accounts for the physical effects of rotation, surface tension and gravity. The effect of the latter is measured by a non-dimensional parameter $0 \leq \delta \ll 1$. We characterise the structure of the steady states depending on the ratio $\ell$ of the cylinder length to its radius. In the absence of gravity ($\delta=0$), in the case $\frac{\ell}{\pi} \notin \mathbb{Z}$, steady states are unique. For $0 < \delta \ll 1$, steady states are shown to be locally unique for any $\ell$. These steady states are stable for $\ell < \pi$, while they are unstable for $\ell > \pi$. Furthermore, in the absence of gravity, for all $\ell > 0$, we show that there exists a manifold of time-periodic solutions. In the critical case $\ell = \pi$, we study the dynamics of the solutions close to the manifold of periodic orbits in the critical case $\ell = \pi$ on the large time scale $\tau = \delta^2 t$. It turns out that in the time scale $\tau$ this dynamics can be approximated by a system of ordinary differential equations.

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel–Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

This paper establishes the global well-posedness of the multi-species Boltzmann equation with large-amplitude initial data in the periodic domain $\mathbb{T}^3$. In contrast to the single-species case, the multi-species mixture model lacks structural symmetry in its collision operators due to the distinct masses of different species. This asymmetry makes it difficult to obtain pointwise estimates for the nonlinear collision terms. Although the Carleman representation for the mixture model introduced in \cite{BD2016} provides a useful reduction of the collision integral, it does not directly yield the desired estimate. To overcome this difficulty, we identify an additional algebraic cancellation structure which leads to the pointwise estimates for the nonlinear terms. By applying this refined approach, we derive the necessary velocity-weighted $L^\infty$ estimates for the nonlinear terms. Furthermore, under the smallness assumption on the initial relative entropy, we establish a uniform lower bound for the nonlinear collision frequency and prove that the large-amplitude solutions exist globally in time and decay exponentially to the global equilibrium.

We study the $\Gamma$-convergence of a class of elastica-type energies defined on immersed planar curves and depending on a small positive parameter $\epsilon$. As $\epsilon\to 0^+$, sequences with equibounded energy develop concentration phenomena in the curvature, leading to the emergence of singularities described by atomic measures. This naturally gives rise to a limiting framework in terms of pointed curves, consisting of a curve together with a measure encoding curvature concentration. We characterize the first-order $\Gamma$-limit in two settings: for immersed open curves with fixed endpoints and boundary conditions on the tangents, and for immersed closed curves of prescribed length. In both cases, the limiting energy depends only on the number of concentration points and is expressed as a sum of contributions, each given by an integer multiple of $2\pi$. A key feature of the problem is that the rescaled energies exhibit a structure closely related to one-dimensional Modica–Mortola type functionals.