偏微分方程分析

We investigate normalized solutions for doubly nonlinear Schr\"odinger equations on the real line with a defocusing standard nonlinearity and a focusing nonlinear point interaction of $\delta$-type at the origin. We provide a complete…

In this paper, we introduce a nonlinear system of partial differential equations, the magnetic two-component Hunter-Saxton system (M2HS). This system is formulated as a magnetic geodesic equation on an infinite-dimensional Lie group…

We prove equivalence between nonnegative distributional solutions of the fractional heat equation and caloric functions, i.e., functions satisfying the mean value property with respect to the space-time isotropic $\alpha$-stable process. We…

We analyze the behavior of an isentropic gas in a narrow pipe with periodically-varying cross-sectional area. Using multiple-scale perturbation theory, we derive homogenized effective equations, which take the form of a constant-coefficient…

We prove the global-time existence of weak solutions to the supercooled Stefan problem. Our result holds in general space dimensions and with a general class of initial data. In addition, our solution is maximal in the sense of a certain…

In geophysical environments, wave motions that are shaped by the action of gravity and global rotation bear the name of gravito-inertial waves. We present a geometrical description of gravito-inertial surface waves, which are low-frequency…

This study investigates the $L^1_{\operatorname{loc}}$ compactness of velocity averages of sequences of solutions $\{u_n\}$ for a class of kinetic equations. The equations are examined within both deterministic and stochastic heterogeneous…

We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimization approach introduced for the local case, we construct…

We study phase transitions for repulsive-attractive mean-field free energies on the circle. For a $\frac{1}{n+1}$-periodic interaction whose Fourier coefficients satisfy a certain decay condition, we prove that the critical coupling…

We show that jets of initial data can be approximated up to arbitrary order by finite-gap solutions for classes of so-called BKM systems of PDEs introduced by Bolsinov--Konyaev--Matveev, which include classical PDEs such as KdV,…

We establish an isoperimetric type inequality for the level sets of functions in fractional Sobolev spaces. This answers a question posed by the first author in a previous paper. To obtain it, we work out a slight modification of some…

In this article, we study Lions' density patch problem in two space dimensions at critical regularity. We prove global existence, uniqueness, and stability for a fluid occupying a bounded Lipschitz region surrounded by vacuum and evolving…

We prove the existence of global weak entropy solutions for a class of non-symmetric Keyfitz-Kranzer type systems that includes lubrication models for thin-film flow. We identify a family of entropy/entropy-flux pairs for these first-order…

In this work, we initiate the study of the biharmonic heat equation in a spatial bounded domain subject to dynamic boundary conditions involving the bi-Laplace-Beltrami operator on the boundary. The boundary heat equation is coupled to the…

In this paper, we investigate the long-time dynamics of a repulsive Keller-Segel chemotaxis system. The model features negative chemotaxis, logistic growth and a cell death term, accounting for a lethal chemorepellent that is self-produced…

We prove scattering for the defocusing energy-critical non-linear wave equation with Dirichlet boundary conditions outside two strictly convex obstacles in dimension three. This is the first large data scattering result for such an equation…

We consider the three-dimensional incompressible MHD system. Any weak solution satisfying a strong energy inequality is $L^2$-asymptotically stable around a Landau solution. Under an additional integrability assumption on the initial…

We prove a quantitative averaging lemma for spatially dependent vector fields. Our proof is based on an iteration of the regularizing operator and some elementary considerations about the local inversion theorem.

This paper is devoted to the study of the compressible boundary layer equations in the Gevrey-2 solution space. Compared to the classical Prandtl equation, the additional complexity arises from the strong interaction between viscous layer…

This paper investigates the dynamics of closed vortex filaments in $\R^3$ governed by the Localized Induction Equation. Recently, Aiki and Higaki (2026) established the nonlinear orbital stability of circular vortex filaments under…