偏微分方程分析

The Casten-Holland and Matano theorem for interior reactions states that no nonconstant stable solutions exist in convex domains $\Omega$ of $\mathbb{R}^n$ under zero Neumann boundary conditions. In this paper we establish that the…

We study dispersive properties of the one-dimensional Schr{\"o}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with…

The classical isocapacitary inequality states that, among all sets of fixed volume, the ball uniquely minimizes the capacity. While this result holds in the continuum, it fails in the discrete setting, where the isocapacitary problem may…

In this short paper, we show that any Lam\'e system whose Dirichlet-to-Neumann map for the elastic wave equation agrees with the one arising from the homogeneous Lam\'e system must actually be homogeneous. We do not need to impose any…

This paper concerns the diffusive limit of the time evolutionary Boltzmann equation in the half space $\mathbb{T}^2\times\mathbb{R}^+$ for a small Knudsen number $\varepsilon>0$. For boundary conditions in the normal direction, it involves…

Rogue waves are extreme ocean events characterized by the sudden formation of anomalously large crests, and remain an important subject of investigation in oceanography and mathematics. A central problem is to quantify the probability of…

A polyatomic ideal gas with weak interaction between the translational and internal modes is considered. For the purpose of describing the behavior of such a gas, a Boltzmann equation is proposed in the form that the collision integral is a…

We address the fluid-structure interaction between a viscous incompressible fluid and an elastic plate forming its moving upper boundary in three dimensions. The fluid is described by the incompressible Navier-Stokes equations with a free…

We study a time-delayed variant of the Hegselmann-Krause opinion formation model featuring a small group of leaders and a large group of non-leaders. In this model, leaders influence all agents but only interact among themselves. At the…

We prove that, modulo rigid motions, the Hopf map is the unique minimizer of the Faddeev--Skyrme energy in its homotopy class, provided that the radius of the target 2-sphere is not smaller than the radius of the domain 3-sphere.

We are concerned with the existence of $T$-periodic solutions to an equation of type $$\left (|u'(t))|^{p(t)-2} u'(t) \right )'+f(u(t))u'(t)+g(u(t))=h(t)\quad \mbox{ in }[0,T]$$ where $p:[0,T]\to(1,\infty)$ with $p(0)=p(T)$ and $h$ are…

This paper investigates steady state solutions of a vasculogenesis model governed by coupled partial differential equations in a bounded two dimensional domain. Explicit steady state solutions are analytically constructed, and their…

Telomeres are repetitive sequences situated at both ends of the chromosomes of eukaryotic cells. At each cell division, they are eroded until they reach a critical length that triggers a state in which the cell stops to divide: the…

We study a generalization of the manifold-valued Rudin-Osher-Fatemi (ROF) model, which involves an initial datum $f$ mapping from a curved compact surface with smooth boundary to a complete, connected and smooth $n$-dimensional Riemannian…

We consider the free boundary problem for a 3-dimensional, incompressible, irrotational liquid drop of nearly spherical shape with capillarity. We study the problem from the beginning, extending some classical results from the flat case…

We derive in this paper Gaussian estimates for a general parabolic equation $u_{t}-\big(a(x)u_{x}\big)_x= r(x)u$ over $\mathbb{R}$. Here $a$ and $r$ are only assumed to be bounded, measurable and $\mathrm{essinf}_{\mathbb{R}} a>0$. We first…

We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Lie group. The corresponding Gibbs measure is given by a Brownian motion on the Lie group, which plays a central role in stochastic geometry. Our main theorem is…

The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work,…

We prove a Hopf bifurcation theorem in general Banach spaces, which improves a classical result by Crandall and Rabinowitz. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular,…

In this paper, we consider a particular class of solutions to the Boltzmann equation which are referred to as homoenergetic solutions. They describe the dynamics of a dilute gas due to collisions and the action of either a shear, a dilation…