Papers

For any $0\le\gamma<1/3$ we construct weak solutions $(v,B,p)$ of the ideal MHD equations with $v,B\in C^\gamma(\mathbb T^3\times\mathbb R)$, which conserve neither the total energy nor the cross-helicity and have nontrivial magnetic helicity. We also establish anisotropic Hölder bounds with distinct velocity and magnetic exponents and stronger regularity along every magnetic field line. The admissible exponents connect the Goldreich–Sridhar \cite{GoldreichSridhar1995} spatial and parallel pair $1/3, 1/2$ with the $1/4$ spatial scaling of the Iroshnikov–Kraichnan weak-turbulence picture \cite{Iroshnikov1963,Kraichnan1965}. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is regarded as a path of volume-preserving diffeomorphisms; the proof is then based on the interplay between classical convex integration techniques and geometric constructions at the level of the Lie algebra of this Lie group. Our work substantially extends the recent result of Enciso, Peñafiel-Tomás and Peralta-Salas \cite{EnPePe} and can be used to reprove \cite{GiRa} of Giri and Radu without a Newton iteration.

We prove the existence of a McKean–Vlasov stochastic process with jumps associated to the nonlinear parabolic equation $\partial_t u = \Delta_p u + \Delta_p^s u$ in $\R^N\times(0,\infty)$, where $\Delta_p$ is the $p$-Laplacian and $\Delta_p^s$ is the fractional $p$-Laplacian. The algorithm used is the following : first, after proving the existence of a solution for the PDE presented earlier, we rewrite it as a nonlinear Fokker-Planck-Kolmogorov equation whose solution-measure is guaranted when $p\ge4$. Then we solve the martingale problem associated to our FPKE via a new nonlinear supersition principle. Finally, thanks to the martingale solution obtained, we derive the existence of a weak solution for the McKean-Vlasov's type SDE with jumps whose infinitesimal generator is a << hybrid version >> of the operator $\Delta_p+\Delta_p^s$.

This work focuses on range characterization of spherical mean transform (SMT) in odd-dimension Euclidean space. In a recent work, a simpler description characterizing the range of SMT in odd dimensions was derived. This involved symmetry relations involving certain linear ordinary differential operators acting on the coefficients of the spherical harmonics expansion of the function in the range. The sufficiency part of this characterization was shown using an existing range characterization result due to Agranovsky-Kuchment-Quinto. In the current work, we provide a direct and constructive proof showing the sufficiency of the aforementioned symmetry conditions and we obtain this without invoking any of the existing range characterization results. As an added advantage, given a function $g$ satisfying the range conditions, our approach provides an explicit construction of a function $f$ such that the SMT of $f$ is $g$.

In [SX25], we introduced the notion of nondegenerate cylindrical singularity using the ideas from dynamical systems. In this paper, we further study the properties of these singularities, showing that they are stable under small perturbations. Furthermore, we show that we can perturb some mean curvature flow with degenerate singularities to produce nondegenerate singularities.

We investigate time-dependent semiclassical measures associated with the magnetic Schrödinger flow on the flat two-dimensional torus in long-time regimes. Under a geometric nonvanishing assumption on an effective magnetic force, we identify two thresholds and prove a dynamical form of quantum unique ergodicity for such systems. If the time scale $\tau_h \gg h^{-1/2}$, then the configuration marginal is the normalized Lebesgue measure on $\mathbb T^2$. In addition, if $\tau_h \gg h^{-1}$, then the momentum marginal is, conditionally on each regular level set, the normalized invariant measure on that curve.

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equation with a nonlinearity combining a logarithmic term and a local nonlinearity which has polynomial growth. Our main result is the construction of solutions in the energy space $W_1(R^d)$ in the subcritical case. We also discuss a sufficient condition for global existence in the $L^2$-critical case in each of these spaces. Moreover, we establish a more general Cauchy theory in the space $\Sigma_\alpha$, as introduced without details in [6].

The present paper studies the initial-boundary value problem of the nonhomogeneous incompressible two-phase magnetohydrodynamic model with Landau potential in a bounded smooth domain in $\mathbb{R}^d$($d = 2, 3$). More precisely, we construct the existence of local in time in three dimension and the global existence of strong solution in two dimension with arbitrary large data and bounded density. In addition, the uniqueness of the solution is proved through the weighted energy estimates, the shift of integrability method and Lagrangian approach.

We study an uncountable system of McKean-Vlasov SDEs with coefficients of Nemytskii-type which are driven by a family of essentially pairwise independent Wiener processes. These SDEs interact through a Graphon kernel by means of their one-dimensional time marginal law densities evaluated in the spatial coordinate. We prove the existence and uniqueness of probabilistically weak solutions to such SDEs under mild conditions on the drift coefficient and the Graphon kernel. Furthermore, we prove that these solutions are, in fact, probabilistically strong by essentially proving a (restricted) Yamada–Watanabe theorem on Fubini extension spaces. For the associated system of nonlinear Fokker-Planck equations, we prove a new uniqueness result where we can allow for density dependent diffusion coefficients of Nemytskii-type.

Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.

We prove the existence of admissible subsolutions to the Dirichlet problem for symmetric augmented $k$-Hessian type equations. An important sufficient condition is the uniform $(k-1)$-$A$-convexity of the domain $\Omega,$ where $A(x, z, p)$ is the augmented symmetric matrix appearing in the equation. This condition was originally introduced by F. Jiang, N. S. Trudinger, and X.-P. Yang and we have chosen a special their case. The structural conditions on the matrix $A(x, z, p)$ include its growth with respect to the variables $z$ and $p,$ particularly requiring that some of its first and second derivatives are sufficiently small in a sufficiently small neighborhood of the boundary. Under certain structural conditions on $A(x, z, p),$ the uniform $(k-1)$-$A$-convexity of $\Omega$ is also a necessary condition for the existence of admissible subsolutions of the equation in a neighborhood of the boundary. Our results extend the classic result by L. Caffarelli, L. Nirenberg, and J. Spruck from the case $A \equiv 0$ to the general case $A \neq 0.$ Our same theorems are valid also for augmented quotient Hessian type equations.

We consider the boundary trace process of a new class of sticky Brownian motions and study the telegraph signals of the boundary local time. For the fractional telegraph equation \begin{align*} (\sigma/\eta) D^\alpha_t v(t,x) + D^{2\alpha}_t v(t,x) = \frac{\partial^2 v}{\partial x^2}(t,x), \quad t>0,\, x \in \mathbb{R}, \quad \alpha \in (0,1] \end{align*} we provide a probabilistic representation of the solution in anisotropic Sobolev spaces and compare it with well-known representations in the literature. We subsequently discuss the associated processes and provide their pathwise representations. Based on these representations, we introduce a characterization of the local times for a wide class of sticky Brownian motions on $\Omega$ governed by the non-local dynamic boundary condition \begin{align*} \eta D^\alpha_t \varpi(t,x) = - \sigma \partial_n \varpi(t,x), \qquad t>0, \; x \in \partial \Omega \end{align*} where $D^\alpha_t$ denotes the fractional derivative in the Caputo-Džrbašjan sense. We focus mainly on the interval $[a,b]$ to construct a prototype model, and then lay the foundation for the analysis on balls in $\mathbb{R}^d$. The smooth interpolation between wave propagation and diffusion under anomalous dynamics captures the behaviour of the underlying sticky Brownian motion experiencing significantly prolonged trapping times on the boundary. The solution $v$ retains its continuity up to $t=0$ in $H^1(\mathbb{R})$, which corresponds to the mean-square continuity of the telegrapher's process at the initial instant. However, for $t>0$, the severe sticky effect causes the stochastic trajectories to undergo prolonged trapping periods, leading to highly irregular and rough paths for the local time of the sticky Brownian motion. In our construction the Brownian structure appears immediately, rather than only as a hydrodynamic limit.

In this paper we address a basic question that has not been considered in the literature on the parabolic Regularity problem. Let $\Omega$ be the region above the graph of a $\mathrm{Lip}(1,\tfrac12)$ function $\phi$ on $\mathbb R^{n-1}\times\mathbb R$. The corresponding $\dot L^p_{1,1/2}$ Regularity boundary space on $\partial\Omega$ is defined by pulling the datum back to the flat space where we have the norm $\|\nabla f\|_{L^p}+\|D^{1/2}_t f\|_{L^p}$. As there is more than one graph parametrisation of such a domain, this could petentially lead to different notions of $\dot L^p_{1,1/2}(\partial\Omega)$. This issue arises only on time-varying domains, which is likely why the issue has not been noticed before. We show by example that the boundary spaces can indeed disagree. We then prove that an additional assumption $D^{1/2}_t\phi\in\mathrm{BMOpar}$ (called the Lewis–Murray condition), resolves the issue completely: the spaces defined through different graphs agree.\medskip This gives us a well-defined Regularity datum space on every Lewis–Murray graph domain. We then extend the duality between the parabolic Dirichlet and Regularity problems, proved on cylinders by the author and E.~Nyström, to such domains.

We consider $m$-dimensional currents in $\mathbb{R}^{m+1}$ that almost minimize an anisotropic energy whose integrand is continuous in the space variable and $C^{2,{\rm Dini}}$ in the normal variable. We prove that if the boundary of such a current is differentiable, then the current admits a flat blowup at every boundary point.

We classify isolated singularities of classical solutions to the capillary equation with negative gravity, \[ \operatorname{div}\frac{Du}{\sqrt{1+|Du|^2}}=-u \qquad\text{in }B_R\setminus\{0\}\subset\mathbb R^n, \qquad n\ge2, \] without a priori assumptions on symmetry, sign, one-sided boundedness, or blow-up rate. Every such solution is either bounded near the puncture or tends uniformly to $+\infty$ or $-\infty$. In the bounded case, the solution extends across the puncture as a $W^{1,1}_{\mathrm{loc}}$ distributional solution; this extension has a unique continuous representative when $2\le n\le7$. In the unbounded case, for some $\varepsilon\in\{-1,1\}$, \[ u(x)=\varepsilon\left(\frac{n-1}{|x|} -\frac{n+3}{2(n-1)^2}|x|^3\right)+O(|x|^5), \] uniformly in the angular variable. In every fixed smaller ball, all sufficiently high level sets of $\varepsilon u$ are connected, smooth, strictly convex hypersurfaces enclosing the puncture. The corresponding pressure-rescaled graphs converge smoothly with multiplicity one to the round cylinder $\partial B_{n-1}\times\mathbb R$. Nonradial examples form infinite-dimensional families that agree with the radial pole to every algebraic order, so the complete asymptotic expansion does not determine the singular solution germ. The proof of the classification combines critical tail estimates and logarithmic $BV$ compactness with level-set rigidity and the translation identities of the capillary equation. The critical two-dimensional case requires an additional finite-height analysis to exclude a translation defect.

It is known that a major difficulty in proving Hölder regularity for solutions to quasilinear singular parabolic equations of $p$-Laplace type via the method of intrinsic scaling is to establish the decay estimate of the space-time measure of level sets. In this paper, we consider an inhomogeneous singular parabolic $p$-Laplace equation with nonnegative time-independent forcing and Dirichlet data. By combining a time-rescaling argument with $L^\infty$ comparison estimates, we derive Lipschitz regularity in time after any fixed positive elapsed time, which reduces the problem to an elliptic-type decay estimate for the spatial measure of level sets that is essentially uniform in time. This reduction enables us to address the above obstacle.

We examine the inverse problem of recovering the potential term of an n-dimensional heat equation from a single time solution profile. We show that it is possible to uniquely determine this potential term via a stability inequality using solution measurements taken at a fixed time, assuming the potential is known within an arbitrary subdomain. The approach is based on the Carleman estimate. Additionally, we discuss the optimality of this type of observation.

In this paper, we introduce a novel multi-fluid approximation scheme for the one-dimensional Vlasov-Poisson system, that is relevant for arbitrarily long time intervals. For low regularity solutions, we establish an error estimate in the Wasserstein-1 distance, obtaining a convergence rate of order $O(h^{\frac{2}{3}})$, where h denotes the velocity grid size. This establishes the consistency, as the grid size tends to zero, between the pressureless Euler-Poisson and the Vlasov-Poisson systems. Numerical illustrations are given to illustrate the efficiency of our approach.

We establish an integral Harnack inequality and expansion of positivity for solutions to fractional porous medium-type equations in the fast diffusion regime, under optimal tail assumptions.

We prove that strong solutions to the incompressible, homogeneous Navier-Stokes system on $\mathbb{R}^3$ are global in time for initial data in the Sobolev space $H^1_\sigma(\mathbb{R}^3)$. The central ingredient of the proof is a new energy estimate that rules out finite-time blow-up in $L^r(\mathbb{R}^3)$ for $r = 2 + 2 / \sqrt{3}$. This estimate is based crucially on properties of the material derivative in combination with the divergence-free constraint, together with an intricate estimate that harnesses the delicate interplay between the viscous coercivity of the new energy functional and the control of a perturbing pressure term. Notably, the estimate does not carry over to systems that merely share the standard energy structure, such as T. Tao's averaged Navier-Stokes system. As a consequence of the new energy estimate, every Leray-Hopf weak solution $u$ to the homogeneous Navier-Stokes system is globally unique whenever the initial value $u(0, \cdot)$ belongs to $L^2_\sigma(\mathbb{R}^3) \cap L^3(\mathbb{R}^3)$. Even for $u(0, \cdot) \in L^2_\sigma(\mathbb{R}^3)$, the only possible non-uniqueness in the class of Leray-Hopf weak solutions is initial branching, and every such solution is $C^\infty$-smooth on $\left( 0, \infty \right) \times \mathbb{R}^3$. Moreover, if $u(0, \cdot)$ is smooth with derivatives of all orders decaying rapidly at infinity, we show that $u$ is in fact $C^\infty$-smooth on all of $\left[ 0, \infty \right) \times \mathbb{R}^3$.

We consider solutions of space-periodic reaction-diffusion-advection equations in the whole space R^N, with general unbounded initial support. We show the existence of spreading sets for the solutions at large time, and variational formulas for the spreading speeds. The location of the upper-level sets of a solution with unbounded initial support is estimated in terms of that initial support and the spreading set for the solutions with bounded initial supports. The results hold for the standard classes of Fisher-KPP, ignition, or bistable reactions. The general results apply to the cases of V-shaped and Lambda-shaped initial supports. The proofs rely in particular on upper estimates of what we call retracting solutions.

We consider a class of neural field models describing the macroscopic activity of an infinite population of neurons organized into finitely many stacked layers, written as a system of semi-linear higher-order parabolic equations. For this class of biologically inspired equations, we show that the higher-order dissipation, modeling gap junction activity regulation, induces a spectral gap ensuring that the associated semigroup possesses an inertial manifold, that is, a finite-dimensional manifold attracting all orbits of the infinite-dimensional system of neural activity. Our main result is an upper bound on the dimension of this manifold, giving its explicit scaling in the dissipation order and strength, the leaking rate, and the norm of the connection operator. We interpret this as a step toward the mathematical modeling of the empirically motivated concept of neural manifold, namely the idea that the collective activity of a large neural population lies on a low-dimensional manifold. For this class of equations, we further prove universal approximation properties, namely stationary pattern expressivity in terms of the input and input-dependent dynamical expressivity over finite time horizons. Finally, we provide sufficient conditions for the existence of traveling waves and nontrivial stationary solutions.

In this paper, we study the Liouville equation with a nonlinear Neumann boundary condition \begin{equation*} \begin{cases} -\Delta u = Ke^{2u} & \text{in } \mathbb{B}^{2},\\[2mm] \dfrac{\partial u}{\partial \nu}+\lambda = ke^{u} & \text{on } \partial\mathbb{B}^{2}, \end{cases} \end{equation*} where $K,k\in\mathbb{R}$, $\lambda\in(0,1]$ are constants, and $\nu$ denotes the outward unit normal on $ \partial\mathbb{B}^{2}$. We establish a classification of all smooth solutions to the equation. Our approach is a boundary-adapted P-function method which treats all signs of $K$ and $k$ within a unified framework. As applications of the classification result, we establish a family of sharp Sobolev-trace-type inequalities encompassing the classical Lebedev–Milin inequality, together with a corresponding deficit estimate.

We prove that the Dirichlet-to-Neumann map uniquely determines a smooth real Schrödinger potential on any prescribed compact connected smooth Riemannian manifold of dimension at least three with nonempty smooth boundary. The proof combines local continuation of Green kernels with uniform estimates for the associated integral representations on shrinking spheres. These estimates yield local recovery of the potential, and a continuation argument extends this recovery throughout the manifold.

We study the stationary focusing nonlinear Schrödinger equation on the product $\mathbb{R}^N \times \mathcal{G}$ of the Euclidean space with a compact metric graph, in the mass-constrained variational setting. Such a product is a hybrid structure of a new type: all its faces are $(N+1)$-dimensional and are glued along interfaces of codimension one, so that the energy space is a genuine Sobolev space, while both the metric and the topology of the graph enter the variational problem. We first develop the functional framework, giving two equivalent descriptions of $H^1(\mathbb{R}^N \times \mathcal{G})$, introducing partial rearrangements in each of the two variables together with the corresponding Pólya–Szegő inequalities, and proving Gagliardo–Nirenberg inequalities in a localized form, with a comparison of the optimal constants with those of $\mathbb{R}^{N+1}$ and of the half-space. We then study the mass-constrained problem. Ground states exist for every mass when $2<p<2_*:=2+4/(N+1)$. At the critical exponent $p=2_*$, they exist below a graph-dependent threshold lying between one half of the Euclidean critical mass and the full Euclidean critical mass. The latter value is attained when the graph admits a cycle covering, whereas the threshold is exactly halved in the presence of a terminal edge. In both cases, the threshold is sharp. For $2_*<p<2+4/N$ global minimizers do not exist, but we prove the existence of local minimizers below a further mass threshold, for which we give an explicit lower bound. Finally, we describe the dimensional crossover: below a critical mass the minimizers do not depend on the graph variable, and we characterize the threshold below which the semi-trivial solution is a local minimizer in terms of the first nonzero eigenvalue of the Kirchhoff Laplacian on $\mathcal{G}$.

In this article, we prove the sharp Strichartz estimates on the one-dimensional torus.

{Let $\lambda_i = \nu_i(F)$ denote the three singular values of the deformation gradient $F \in {\rm GL}^+(3)$.} We consider the family of incompressible isotropic energies $ W_\psi(F)=\sum_i \psi\left(|\!\log\lambda_i|\right)$ with $\psi:[0,\infty)\mapsto\mathbb{R}$. Set $g(s)=\psi\left({\rm arcosh}\frac{s}{2}\right)$ for all $s\geq2$. If $g$ has a convex and non-decreasing extension $\bar g$ to $[0,\infty)$, then $W_\psi$ is the restriction to ${\rm SL}(3)$ of the explicit polyconvex function \ {$ F\mapsto\sum_i \bar g\left(\nu_i(F+{\rm Cof} F)\right). $} The proof uses the identity $\nu_i(F+{\rm Cof} F)=\lambda_i+\lambda_i^{-1}$, up to permutation, on ${\rm SL}(3)$ and Ball's convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in ${\rm SL}(3)$ and derive a convenient one-dimensional differential sufficient condition. In particular, $ F\mapsto\sum_i e^{\log^2\!\lambda_i} $ is rank-one convex on ${\rm SL}(3)$ and possesses the stated polyconvex extension. A simple-shear computation shows that scalar convexity in $\log \lambda_i$ alone is insufficient; the quadratic Hencky energy $\sum_i \log^2\lambda_i$ is not rank-one convex on ${\rm SL}(3)$.

In this paper, we study the nonlinear Schrödinger equation with a nonlinearity combining a focusing logarithmic term and a smooth $L^2$-subcritical function. Our main result is the construction of multisoliton solutions for this equation. These special configurations behave asymptotically, for large times, as the superposition of several solitons moving with distinct velocities. The proof relies on an iterative construction method combined with localized energy estimates adapted to the mixed structure of the nonlinearity.

The present article investigates the well–posedness of a discrete coagulation–annihilation model originally introduced by E. Ben-Naim and P. Krapivsky in Physical Review E (1995). We first establish the existence of solutions for a broad class of coagulation and annihilation rates, including rates that may be unbounded. These results are obtained by combining uniform estimates with Helly's selection theorem and the Rellich–Kondrachov compact embedding theorem. Under appropriate structural assumptions on these rates, we further prove uniqueness and continuous dependence of solutions on the initial data, thereby establishing well–posedness of the model. Finally, for a specific class of coagulation and annihilation rates, we additionally study the differentiability of solutions and characterize their long–time asymptotic behaviour. These results provide a rigorous mathematical framework for the analysis of the model and contribute to a deeper understanding of its well–posedness, regularity, and long-time dynamics.

We introduce centro-sectional measures with parameters q,m for log-concave functions on Rn, defined in terms of the q-th moments of their Radon transforms with respect to the Haar measure on m-dimensional subspaces, where m=1,...,n-1, and establish the corresponding variational formulas. Our measures generalize the notion of dual curvature measure if q=1, and are related to the Sine transform if q=2. In line with the coarea formula for log-concave functions as BV functions, the variational formulas give rise to the Euclidean centro-sectional measures and the spherical centro-sectional measures. In the symmetric setting, we solve the associated even functional centro-sectional Minkowski problem, which asks which pairs of measures can arise as the centro-sectional measures of an even log-concave function.

For every sufficiently large N, we prove that the regular N-gon uniquely minimizes the first Dirichlet eigenvalue among convex polygons of prescribed area with at most N sides, up to rigid motions. Quantitative Faber–Krahn stability localizes minimizers near the disk but leaves their discrete geometry undetermined. We encode that geometry by a centered measure of exterior angles and prove an inverse-cubic Fourier inequality whose equality case is the uniform root configuration. Its defect gives a lower bound for the quadratic spectral excess over the regular polygon. Cyclic cancellation and a common material-coordinate comparison make the nonlinear difference small relative to the same defect. Exact constraint coordinates extend this comparison to arbitrarily small positive defects. Minimality then forces zero defect, which characterizes the regular polygon.