偏微分方程分析

In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial…

We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for…

We study the plastic Burgers equation in one space dimension, i.e., the Burgers equation featuring an additional term formally given by the p-Laplacian with p=1, or rather, by the multivalued subdifferential of the total variation…

This article aims to study the long-time dynamics of the linear viscoelastic plate equation $\displaystyle{u_{tt}+\Delta^2 u-\int_{\tau}^tg(t-s)\Delta^2u(s)ds=0}$ subject to nonlinear and nonlocal boundary conditions. This model, with…

We study the large-population limit of interacting particle systems evolving on adaptive dynamical networks, motivated in particular by models of opinion dynamics. In such systems, agents interact through weighted graphs whose structure…

We study the time-harmonic scattering by a heterogeneous object covered with a thin layer of randomly distributed sound-soft nanoparticles. The size of the particles, their distance between each other and the layer's thickness are all of…

In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-\Delta)_p^s u_\lambda=\lambda u_\lambda^q - b(x)u_\lambda^r \quad \text{in} \;\Omega,\\…

For any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type~I blow-up of the $L^\infty$ norm at time $T_*>0$, while remaining smooth in space and time on $\mathbb…

Time-frequency representations stemmed in 1932 with the introduction of the Wigner distribution. For most of the 20th century, research in this area primarily focused on defining joint probability distributions for position and momentum in…

We consider an optimal control problem for a two-dimensional Navier-Stokes-Cahn-Hilliard system arising in the modeling of fluid-membrane interaction. The fluid dynamics is governed by the incompressible Navier-Stokes equations, which are…

In this paper we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and…

We investigate the long-time behavior of a nonlocal Cahn-Hilliard equation in a bounded domain $\Omega\subset\mathbb{R}^d$ $(d\in\{2,3\})$, subject to a kinetic rate-dependent nonlocal dynamic boundary condition. The kinetic rate $1/L$,…

The main objective of this manuscript is to investigate the global behavior of the solutions to the viscoelastic wave equation with a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as a…

We study a mean field planning problem in which the initial density is a Dirac mass. We show that there exists a unique solution which converges to a self-similar profile as time tends to $0$. We proceed by studying a continuous rescaling…

We study a thermodynamically consistent diffuse interface model that describes the motion of a two-phase flow of two viscous incompressible Newtonian fluids with unmatched densities and a soluble surfactant in a bounded domain of two or…

The introduction of abstract Friedrichs operators in 2007-an operator-theoretic framework for studying classical Friedrichs operators has led to significant developments in the field, including results on well-posedness, multiplicity, and…

Classical results from Sturm-Liouville theory establish that the Morse index of a one-dimensional Sturm-Liouville operator defined on $\mathbb{R}$ is equal to the number of its associated conjugate points. Recent advancements by Beck et…

We concern the long-time behavior of mild solutions to the spatially homogeneous Boltzmann--Fermi--Dirac equation with moderately soft potential. Based on the well-posedness results in [X-G. Lu, J. Stat. Phys., 105, (2001), 353-388], we…

We study the two dimensional viscous Boussinesq equations, which model stratified flows in a circular domain under the influence of a general gravitational potential $f$. First, we show that the Boussinesq equations admit steady-state…

We consider radially symmetric solutions of the degenerate Keller-Segel system \begin{align*} \begin{cases} \partial_t u=\nabla\cdot (u^{m-1}\nabla u - u\nabla v),\\ 0=\Delta v -\mu +u,\quad\mu =\frac{1}{|\Omega|}\int_\Omega u, \end{cases}…