偏微分方程分析

In this paper, we revisit self-similar solutions of the two-dimensional stationary incompressible Navier-Stokes equations under scaling symmetries, also known as Jeffery-Hamel solutions. We investigate the local patterns of smooth…

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any…

We analyze the sensitivity of solutions to the Fokker-Planck equation with respect to some unknown parameter. Our main result is to provide quantitative upper bounds for the $p$-Wasserstein distance $\mathcal{W}_p$ between two solutions…

A well-known conjecture of De Giorgi -- motivated by analogy with the Bernstein problem for minimal surfaces -- asserts the rigidity of monotone solutions to the Allen--Cahn equation in $\mathbb{R}^{d+1}$, with $d\leq 7$. We establish close…

In this paper, we investigate the incompressible Navier-Stokes equations coupled with the Vlasov-Fokker-Planck equation, which describes a two-phase mixture of the viscous incompressible fluid with particles or bubbles through a frictional…

In this paper, we prove a higher integrability result for very weak solutions of higher-order elliptic systems involving a double phase operator as the principal part. As a model case, we consider \begin{equation} \int_{\Omega} \left( |D^m…

The Neumann--Poincar\'{e} (NP) operator, a fundamental operator in potential theory, has attracted renewed attention for its central role in the analysis of surface plasmon resonances (SPRs). SPRs, characterized by non-radiative…

We study a nonlinear and nonlocal elliptic equation posed on the flat torus. While constant solutions always exist, we show that uniqueness fails in general. Using spectral analysis and the Crandall--Rabinowitz bifurcation theorem, we prove…

We study the focusing power nonlinear wave equation with any power, in Minkowski space of any spacetime dimension. We present a complete understanding of the local stability and scattering theory (both in high regularity spaces) for…

We study higher regularity for weak solutions of the $p$-Laplace equation $-\Delta_p u = f$ in a domain $\Omega \subset \mathbb{R}^n$ for $p$ sufficiently close to 2. For $m \ge 3$, assuming that $f$ satisfies suitable Sobolev and H\"older…

We prove local well-posedness and finite-time blow-up for a restricted fourth-order Prandtl equation posed on the half-line with clamped boundary conditions. The equation arises from a two-dimensional fourth-order Prandtl system via an…

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\Omega\subset\mathbb{R}^3$. We assume that $v_r$, $v_\varphi$, $\omega_\varphi$ vanish on the lateral part of boundary $\partial\Omega$ of the cylinder, and that…

Assume $\alpha\in (0, 2)$ and $d\ge 2$. Let $\mathcal L^\alpha$ be the generator of a symmetric, but not necessarily isotropic, $\alpha$-stable process $X$ in $\mathbb R^d$ whose L\'evy density is comparable with that of an isotropic…

Let $k\ge1$ be a positive integer and let $P_g$ be the GJMS operator $P_{g}$ of order $2k$ on a closed Riemannian manifold $(M,g)$ of dimension $n>2k$. We investigate the compactness of the set of conformal metrics to $g$ with prescribed…

In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy…

For a time-independent potential $q\in L^\infty$, consider the source-to-solution operator that maps a source $f$ to the solution $u=u(t,x)$ of $(\Box+q)u=f$ in Euclidean space with an obstacle, where we impose on $u$ vanishing Cauchy data…

We provide a first-order homogenization result for quadratic functionals. In particular, we identify the scaling of the energy and the explicit form of the limiting functional in terms of the first-order correctors. The main novelty of the…

We study the boundary value problem for nonlinear fourth-order partial differential equation with jumping nonlinearity which can serve, e.g., as a model of an asymmetrically supported bending beam. We focus on a special type of solutions,…

We study the admissible values of the wave speed $c$ for which the beam equation with jumping nonlinearity possesses a travelling wave solution. In contrast to previously studied problems modelling suspension bridges, the presence of the…

We prove that the power-law vortex $\overline{\omega}(x) = \beta |x|^{-\alpha}$, which explicitly solves the stationary unforced incompressible Euler equations in $\mathbb{R}^2$ in both physical and self-similar coordinates, is…