Papers

This work investigates a Hénon-type equation with the cap-shaped weight $V_{\alpha}(|x|)=(4|x|(1-|x|))^\alpha$, which concentrates on the sphere of radius $1/2$ as $\alpha\to\infty$. Particular attention is devoted to ground state solutions and the phenomenon of symmetry breaking in the large-$\alpha$ regime. A complete description of the asymptotic behavior of ground state radial solutions is obtained as $\alpha\to\infty$. The analytical results are further supported by numerical simulations of the radial ground states.

In this note, we introduce an enhanced notion of probabilistic well-posedness for dispersive PDEs with random initial data by imposing stability at the origin as the amplitude of randomization tends to $0$. We then use this notion to re-interpret recent works on "beyond variance blowup" for dispersive PDEs, by the authors with their collaborators (2025, 2026), as probabilistic ill-posedness results. By drawing an analogy to the failure of $C^k$-smoothness of a solution map in the deterministic setting, we interpret variance blowup results as mild probabilistic ill-posedness.

We study well-posedness issues of the stochastic Korteweg-de Vries equation (SKdV) with an additive noise, posed on the real line. By using the Fourier restriction norm method adapted to the Fourier-Lebesgue space in time, we first prove global well-posedness of SKdV in $L^2(\mathbb R)$ without assuming the homogenous Sobolev regularity, which was imposed in a work by de Bouard, Debussche, and Tsutsumi (1999). Then, by adapting the argument by Zhou (1997) to the stochastic setting, we prove optimal pathwise unconditional uniqueness for SKdV in $L^2(\mathbb R)$. In the appendix, we present a short argument for proving boundedness of the multiplication by a sharp cutoff function in the Fourier-Lebesgue and Sobolev spaces, which is of interest in its own right.

We investigate the pathwise well-posedness issue of the stochastic nonlinear wave equation (SNLW) with a multiplicative noise. While the Ito solution theory (= random field solution theory) was established in the '80s, its pathwise well-posedness has remained a challenging open problem for over forty years. By building a unified framework for the Fourier restriction norm method adapted to the $U^p$- and $V^p$-spaces, due to Koch and Tataru (2007), and the Young/rough integration theory via the sewing lemma and controlled paths due to Gubinelli (2004) along with the random tensor estimate for multiple stochastic integrals with respect to (fractional) Brownian motions, we establish pathwise local well-posedness of SNLW in optimal regularity ranges. In particular, in the one-dimensional case with a white-in-time noise, our result covers the case of an almost space-time white noise, which is optimal within the framework of one-parameter rough paths.

We prove the existence of time-periodic solutions of the two-dimensional incompressible Euler equations bifurcating from a translating vortex pair. The reference configuration consists of two symmetric vortex patches of equal strength and opposite sign traveling at constant speed. In a regime of large separation between the vortices, the dynamics may be viewed as a small perturbation of an integrable system. Working in a co-moving frame and using the contour dynamics formulation, we reduce the problem to a nonlinear transport equation for the vortex boundaries. The linearized operator exhibits degeneracies associated with symmetries and transport effects. By combining a Lyapunov-Schmidt reduction, Nash-Moser scheme, and spectral analysis with sharp asymptotic expansions of the eigenvalues in order to overcome the degeneracy, we construct families of non-rigid, time-periodic vortex patch solutions for a large Cantor set of parameters. The analysis reveals that translating dipoles possess a surprisingly rich nearby dynamics, far beyond the classical rigid paradigm usually associated with vortex patch motion. More generally, the approach developed in this work is flexible and robust, and is expected to extend to a broader class of nonlocal PDEs from Fluid Mechanics with degenerate behaviours.

In this note, we study degenerate complex Monge-Ampère equations on singular Stein spaces with right-hand sides depending on the unknown. First, we prove continuity up to the boundary for solutions. Next, under a holomorphic peak-set condition, we prove local Hölder estimates on the regular locus, allowing the boundary datum to fail to be Hölder continuous at the singular boundary. This includes the case of finite singular boundary sets.

We study non-exchangeable interacting diffusions with pairwise interaction strengths encoded by a sequence of matrices. Under suitable structural and denseness conditions on these matrices, we prove a universal Central Limit Theorem for the global fluctuation field. As the number of particles $n$ becomes large, it converges in distribution to the unique solution of a stochastic partial differential equation (SPDE), the same Gaussian limit as in the exchangeable mean field case. The result applies, for instance, to scaled adjacency matrices of $m_n$-regular graphs when $m_n/\sqrt{n}\to\infty$. A spatial interaction model shows that the $n^{-1/2}$ denseness threshold is sharp. The proof proceeds with an analysis in negative Sobolev spaces, building on sharp quantitative propagation of chaos results together with functional inequalities.

In this paper, we prove the existence of $\mathbf{m}$-fold doubly-connected stationary vortex patches for the quasi-geostrophic shallow-water equations. The solutions are obtained through a bifurcation analysis based on the Crandall-Rabinowitz theorem, with either the inner radius of an annulus or the Rossby deformation length serving as the bifurcation parameter. A central feature of the work is the highly nontrivial analysis of modified Bessel functions arising in the spectral study of the linearized operator. The proof requires delicate and extensive manipulations of these special functions, including precise asymptotic expansions, differentiation formulas, recurrence identities, monotonicity properties and the analysis of singular quantities governing the bifurcation mechanism. These ingredients are essential for characterizing the bifurcation points and establishing the transversality conditions. Finally, we investigate the radial symmetry of stationary and uniformly rotating simply-connected vortex patch solutions, therefore motivating the previous bifurcation results.

Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics, and a natural safety question is to bound the probability $\nu_t(\mathcal{A}_H) = \mathbb{P}(Q_t \in \mathcal{A}_H)$ that the trajectory lies in a designated failure region $\mathcal{A}_H$. We study this for a smooth, strongly convex loss in $d$ dimensions, with $\mathcal{A}_H$ separated from the minimizer by an energy gap. At the end of training, the equilibrium mass $\pi(\mathcal{A}_H)$ is exponentially small in $d$, with a complementary energy-barrier rate when the noise is small. Along the trajectory, a shape-free bound $\nu_t(\mathcal{A}_H) \le \pi(\mathcal{A}_H)(1 + \sqrt{\chi_0^2/\pi(\mathcal{A}_H)}\,e^{-mt})$ shows the in-set probability relaxes to (twice) the static value after a burn-in of order $d$, using only the global spectral gap $m$. A worked Ornstein-Uhlenbeck example shows this burn-in is necessary: an angular slice of the equilibrium shell can transiently swell by a factor exponential in $d$, though its equilibrium mass is tiny. To rule this out we introduce a local relaxation rate, defined through the spectral measure of the region's centered indicator rather than a Dirichlet-form Rayleigh quotient. For geometrically isolated regions this rate exceeds the global one, shrinking the burn-in, and with a maximum-principle ceiling it caps the trajectory probability uniformly in time. Strong convexity sets how fast training relaxes, but the shape of the unsafe set decides whether the trajectory bulges through it on the way to equilibrium.

We study quantitative convergence rates of the vanishing viscosity approximation of first-order time-dependent Mean Field Games with regularizing coupling acting in the Hamilton-Jacobi equation. Under standard structural assumptions on the Hamiltonian ensuring convergence of the vanishing viscosity approximation, previous results provide either qualitative convergence for the two unknown of the system or quantitative estimates merely for solutions of the Hamilton-Jacobi equation. In this work, we improve these convergence rates under the same assumptions and establish, in addition, quantitative estimates for the convergence of the associated forward Fokker-Planck equation. As a consequence, we obtain quantitative convergence rates for the full Mean Field Game system, thus extending and strengthening the existing theory for the first-order limit.

We study inverse scattering from conformal infinity for impenetrable topological defects whose interaction surfaces are totally geodesic in hyperbolic space \(\mathbb H^n\), \(n \geq 2\). For a fixed spectral parameter \(\lambda_0 > 0\), the prescribed inputs are boundary labels \(\xi \in \partial_\infty \mathbb H^n\). Each label selects an incoming Helgason mode in the hyperbolic interior. Given a defect \(\mathcal P \Subset \mathbb H^n\), this mode generally fails to satisfy the homogeneous trace condition on the interaction surface and hence generates an outgoing correction. The leading coefficient of this correction at conformal infinity defines the measured far-field pattern. The central question is whether \(\mathcal P\) can be recovered from the far-field patterns corresponding to one or finitely many prescribed boundary labels. This gives a formally determined inverse problem at one fixed spectral parameter. Our first main result establishes the unique determination of totally geodesic defects, an admissible class that includes both bulk components and hypersurface-supported ones. For Dirichlet-type defects, a single boundary label suffices. For Neumann-type defects, \(n+1\) boundary labels are sufficient and in general necessary. These labels are required to satisfy the natural affine-independence condition at conformal infinity. Our second main result provides quantitative stability estimates within the same framework. The hyperbolic Hausdorff distance between two defects is controlled by the discrepancy of their far-field patterns at conformal infinity. The proof combines continuation from conformal infinity with quantitative geodesic reflection across totally geodesic hypersurfaces. Taken together, these results yield a qualitative and quantitative far-field inverse scattering theory at conformal infinity, based on formally determined data.

We study de Rham functional equations driven by two increasing fractional linear transformations. Our main purpose is to relate the singularity theory of the associated solutions to dyadic potential theory on the binary tree. We first prove an existence and uniqueness theorem for increasing, left-continuous solutions in the full range of linear fractional data, and identify the trapping region in parameter space where the solution is continuous. For a large class of parameters we show that the de Rham solution is the normalized cumulative capacitary function of a multiplicative dyadic capacity. This gives a potential-theoretic model for Möbius de Rham systems. We then sharpen Okamura's Hausdorff-dimensional estimates for the singular measure associated with the solution by replacing Hausdorff dimension with dyadic Riesz capacities at the upper endpoint of Okamura's theorem.

We consider Schrödinger operators with purely imaginary potential $P = - h^{2} \Delta + i V ( x )$ on a bounded domain. Assuming that near its critical points the potential $V$ can be approximated by an homogeneous polynomial, we show that in the limit $h \to 0$ the leftmost eigenvalues of $P$ are asymptotically given by the local model associated to the most degenerated critical points of $V$. We give applications of this result to the associated evolution problem including shear flows in fluid mechanics.

This paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables, and by virtue of these estimates, we establish conditional stability estimates for this problem for the first time. Building upon this, we propose a Tikhonov-type regularization functional and derive its associated adjoint system. Furthermore, leveraging the established conditional stability results, we derive the convergence rate of the proposed regularization approach. Finally, we validate the effectiveness of our theoretical findings through extensive numerical experiments.

We prove global higher integrability of the gradient for solutions to nonlinear PDEs with homogeneous Dirichlet condition. We work with general open sets (i.e. not necessarily bounded), under a minimal thickness assumption of the complement. The main focus is on improving the global summability of the gradient, by preserving the “zero trace" Sobolev class. We pay due attention to providing precise a priori estimates. As an application, we get universal global Hölder estimates for solutions of the super-homogeneous Lane-Emden equation for the $N-$Laplacian, in contractible open sets with finite inradius.

We consider the first eigenvalue, $\lambda_\beta(\Omega,A)$, of a two-phase eigenvalue problem for the Laplacian with Robin boundary conditions, where the two phases are characterised by different ellipticity constants. We characterise the conditions under which a ball $B_R$ is a local minimum under a volume constraint for the minimisation problem $A\mapsto\lambda_\beta(B_r,A)$, in terms of the principal Neumann eigenvalue of the fixed inner ball $B_r$.

For radial initial data, we construct explicit higher-order \(L^p(\mathbb R^N)\)-asymptotic profiles for the heat equation with Hardy potential. These profiles, denoted $A_n$ are obtained from the small-argument expansion, up to an arbitrary order \(n\), of the modified Bessel function appearing in the radial Hardy heat kernel. If $u$ is the mild solution generated by this kernel, we prove that the corresponding remainder $u(x,t)-A_n(x,t)$ admits a polynomial decay depending on $n$ in \(L^p(\mathbb R^N)\) as \(t\to\infty\). We also treat the non-radial case through spherical harmonics: each angular mode evolves according to a radial Hardy heat equation with a modified parameter, leading to finite and infinite angular expansion versions of the asymptotic profile under suitable summability assumptions.

In this paper we investigate the existence and qualitative properties of weak solutions for a class of nonlinear fractional equations driven by the spectral fractional Laplacian under mixed Dirichlet–Neumann boundary conditions. More precisely, we consider subcritical problems of the form \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^s u=\lambda u + \mu f(x,u) & \text{ in } \Omega\smallskip, \\ \displaystyle u\chi_{\SD} +\frac{\partial u}{\partial\nu}\chi_{\SN}=0 & \text{ on } \partial\Omega, \end{array} \right. \end{equation*} where \(s\in(1/2,1)\), \(\Omega\subset\mathbb R^N\) is a bounded smooth domain, and the boundary \(\partial\Omega\) is endowed with a mixed Dirichlet–Neumann configuration satisfying suitable geometric and analytic assumptions. Working in the natural spectral fractional framework associated with the mixed boundary operator, we establish the existence of nontrivial weak solutions by means of suitable variational methods. The analysis is based on a topological and variational principle combined with a localization argument involving certain compactly supported cutoff functions. More precisely, the proof relies on a delicate local comparison argument involving localized plateau-type test functions and explicit asymptotic estimates near the origin. Such an approach is sufficiently flexible and may potentially be adapted to several different settings. As applications, we establish simplified existence criteria in the autonomous case through the use of the Chebyshev radius of the domain and present concrete examples illustrating the applicability of the abstract theory.

We study the quadratic nonlinear Schrödinger equation in energy-critical and energy-supercritical regimes and establish unconditional uniqueness at critical regularity on both $\mathbb{T}^{d}$ and $\mathbb{R}^{d}$. We introduce a new infinite quadratic hierarchy that featuring a linear structure for tensor product forms, instead of marginal densities. Consequently, this newly constructed hierarchy differs from the quantum Gross-Pitaevskii hierarchy, and its structure is in fact closer to that of the classical Boltzmann hierarchy. We prove this quadratic hierarchy admits combinatorial structures that are compatible with the bilinear $U$-$V$ estimates we prove for the quadratic nonlinearity. These tools enable us to establish unconditional uniqueness at the critical regularity via a quadratic hierarchy approach, and thus to provide an affirmative answer to Bourgain's uniqueness concern [3,p.152] for the quadratic nonlinearity in the weak-form bilinear case.

We study the fractional Plateau problem for $s$-minimal sets with prescribed exterior datum. For fixed exterior data, minimizers need not be unique, and singularities may occur beyond the critical dimension. We first prove a generic uniqueness theorem: along any strictly increasing family of exterior data, nonuniqueness occurs for at most countably many parameters. We then show that one can make arbitrarily small perturbations for which the interior regularity theory improves by one dimension.

In this work, we investigate the formation of naked singularities for the $3+1$-dimensional Einstein-scalar field system without symmetry assumptions. We generalize the spherically symmetric and self-similar naked-singularity solution constructed by Christodoulou in [4] by prescribing non-spherically symmetric initial data along both incoming and outgoing initial null hypersurfaces. We then establish global existence for the resulting solutions and analyze the singular structure of the inner Cauchy horizon. Our construction is based on employing a notion of four-type differences and designing a system of scale-invariant weighted norms to control the corresponding geometry. We show that the constructed spacetimes retain a global naked-singularity structure, characterized by an incomplete future null infinity and a singular inner Cauchy horizon. Moreover, we derive detailed asymptotics near the inner Cauchy horizon and prove the desired $C^{1, \frac{\kappa}{1-\kappa}+}$ inextendibility of these solutions, where $\kappa\in (0,1/3)$ is the self-similar parameter. This indicates a connection between weak and strong cosmic censorship: for the class of non-spherically symmetric solutions constructed here, the failure of weak cosmic censorship in its strict formulation is accompanied by a quantitative inextendibility mechanism at the inner Cauchy horizon.

The quasilinear Keller-Segel system with flux limitation and indirect signal production u_t=\nabla\cdot\left(D(u)\nabla u\right) -\nabla\cdot\left(u(1+\left|\nabla v\right|^2)^\sigma\nabla v\right), &x\in\Omega, t>0, \\ 0=\Delta v-v+w,x\in\Omega, t>0, \\ w_t=-w+u,x\in\Omega, t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain \Omega\subsetR^N is considered, where D(u)\simeq u^{m-1} as u\simeq\infty. We conclude that For N=1 and any \sigma\in\mathbb{R}, if m\geq0, the classical solution exists globally, and it is moreover bounded if m>0. However, if m<0 and \Omega is a ball, there exist radially symmetric initial data such that the classical solution exhibits finite-time blow-up. For any N\geq2 and m>1-\frac{1}{N}, if \sigma\leq \frac{mN+2-2N}{2N-2}, the classical solution is global. Furthermore, if \sigma<\frac{mN+2-2N}{2N-2}, the corresponding solution is uniformly bounded. For any N\geq2 and m<2-\frac{2}{N}, if \sigma>\max\left\{\frac{N}{2-2N},\frac{mN+2-2N}{2N-2}\right\} and \Omega is a ball, there exist radially symmetric initial data such that the classical solution blows up in finite time.

We study the one-dimensional Gross-Pitaevskii equation with a traveling delta potential and non-zero conditions at infinity. This model describes the effect of a moving impurity in a quantum fluid. Firstly, we show that the associated Cauchy problem is globally well-posed in the energy space. This requires the definition of a conserved energy, which involves the notion of renormalized momentum. Secondly, we study the existence and stability of stationary states in a co-moving reference frame. It is known that there exists an impurity-dependent critical velocity above which no stationary state exists. For velocities below the critical one, two different stationary states appear. We show the orbital stability of the one with higher minimal density.

This paper is concerned with the zero-mass $(p,q)$-Laplacian equation $$ -\Delta_p u-\Delta_q u = |u|^{r-2}u+\lambda |u|^{s-2}u \quad \text{in } \mathbb{R}^N, $$ where $1<q<p<N$. The exponents $r$ and $s$ may belong to either the subcritical or the supercritical range with respect to the critical Sobolev exponents $p^*$ and $q^*$. We establish existence and nonexistence results and show that the sign of the parameter $\lambda$ determines the solvability regimes of the equation. The existence proofs rely on variational methods, truncation arguments, and regularity theory, while the nonexistence results are derived from a suitable Pohozaev-type identity for the $(p,q)$-Laplacian operator.

We consider the linear stability of three-dimensional sessile drops with a free contact line. The equilibrium surface is an axisymmetric solution of the Young–Laplace equation with gravity, fixed volume, and prescribed contact angle. We derive the constrained second variation of the gravity–capillary energy and formulate the associated Jacobi problem. Although variational stability gives nonnegativity, the second variation is necessarily degenerate because horizontal translations preserve the energy. Our main result identifies this degeneracy completely. Under the pressure–volume nondegeneracy condition $(dV/d\lambda\neq 0)$, we prove that the kernel of the constrained Jacobi operator is exactly the two-dimensional space generated by horizontal translations. The proof combines the geometric structure of the Jacobi operator with a Fourier-mode analysis: the axisymmetric mode is ruled out by the pressure–volume condition, the first mode gives translations, and all higher modes are excluded by comparison. This provides the precise linear nondegeneracy underlying stability of droplet dynamics modulo translations.

We establish the $C^1$-regularity of solutions to the obstacle problems associated with $p$-Laplacian type equations, where $1<p<\infty$. Specifically, we prove that the gradient of the solution is continuous under a Dini mean oscillation ($\mathsf{DMO}$) type condition on the data, which includes the coefficient matrix, the source term, and the obstacle function. This result relaxes the classical Dini continuity assumption on the data to a more general mean oscillation condition.

This paper investigates the robustness of quantum observability on compact hyperbolic surfaces under $L^2$ potential perturbations. Since $L^2$ regularity is strictly subcritical in dimension two, for every non-empty open set $\Omega\subset M$ and $T>0$, the solution of $(i\partial_t+\Delta-V)u=0$ satisfies the space-time observability estimate $$ \|u_0\|_{L^2(M)}^2 \leq C\int_0^T \|e^{-it(-\Delta+V)}u_0\|_{L^2(\Omega)}^2\,dt. $$ Our results provide a quantitative confirmation that the delocalization of high-energy quantum states on negatively curved manifolds is robust against $L^2$-class microscopic scattering. The proof combines the hyperbolic dynamics of the geodesic flow with semiclassical analysis. A key ingredient is an $L^4$ spectral cluster estimate with an arbitrarily small loss, obtained by exploiting Jacobi field analysis and Bourgain–Demeter $l^2$-decoupling. This estimate allows us to obtain refined spectral localized semiclassical Strichartz estimates adapted to rough potentials and to show that the potential's contribution vanishes in the propagation of semiclassical measures. The full-support property of invariant semiclassical measures on hyperbolic surfaces then yields the desired observability by contradiction. By the Hilbert Uniqueness Method, the corresponding internal controllability result follows.

For a smooth, complete and normal metric $g = e^{2u}|dx|^2$ with finite total $n$-th order $Q$-curvature on $\mathbb{R}^n$ with dimension $n \geq 2$, we first show that everywhere non-negativity (resp. non-positivity) $n$-th order $Q$-curvature $Q_g^{(n)}$ implies everywhere non-negativity (resp. non-positivity) of the sectional curvature. Based on this fact, we secondly show that, once $Q_g^{(n)}$ is non-negative, then for any compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial\Omega$, the following sharp isoperimetric inequality holds: $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}} \left(1 - \frac{2}{(n-1)!\,|\mathbb{S}^n|} \int_{\mathbb{R}^n} Q_g^{(n)} \, d\mu_g\right) |\Omega|_g.$$ The third claim in this article is that, if the $n$-th order $Q$-curvature, $Q_g^{(n)}$, is non-positive and under the main assumption that Cartan-Hadamard conjecture holds true, then we have the sharp inequality $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}}|\Omega|_g.$$

We study the nonlinear stability of an equilibrium with a background magnetic field for the three-dimensional compressible magnetohydrodynamic (MHD) equations in the whole space $\mathbb{R}^3$, in the strongly anisotropic regime where the velocity is dissipated only in the horizontal directions and the magnetic field is diffused in a single direction. We prove that for initial data sufficiently close to the equilibrium in a Sobolev space, the system admits a unique global-in-time solution that remains close to the equilibrium and enjoys quantitative dissipation estimates. The proof overcomes the severe lack of dissipation through two mechanisms: the background magnetic field is shown to generate enhanced dissipation for the magnetic field and the density, while a nonlinear cancellation mechanism is devised to resolve the loss of vertical derivatives caused by the compressible coupling.

In this paper, we establish the power convexity of solutions to the complex Monge-Ampère equation $\det(u_{i\overline{j}})=1$ in a convex set in the complex 2 dimensional Euclidean space. Even the convexity result is only for 2 dimension, our approach is applicable to all dimensions.