偏微分方程分析

This paper focuses on Cauchy problem for the three-dimensional two-fluid type model, in which the presence of vacuum is permitted. Under some assumptions that the initial data satisfy appropriate regularity conditions and a compatibility…

This paper is concerned with space inhomogeneous quantum Fokker-Planck equations posed on a classical kinetic phase space. The nonlinear factor $f(1\pm f)$ appears both in the transport term and in the collison part of the Fokker-Planck…

We consider fractional variants of divergence form problems with highly oscillatory local coefficients. We characterise the convergence of these coefficients by means of classical $H$-convergence covering the local behaviour of the…

We study the acoustic operator $A_{v,\rho }:=v^{2}\rho\nabla\!\cdot\rho^{-1}\nabla$ with transmission conditions at the boundary of $\Omega=\Omega_{1}\cup\dots\cup\Omega_{n}$, where the $\Omega_{\ell}$'s are connected disjoint open bounded…

We derive an explicit inversion algorithm for the spherical Radon transform in odd dimensions with partial radial data. We prove that the reconstruction of the unknown function can be reduced to solving ordinary differential equations,…

This article studies the Cauchy problem for the scalar conservation law \[ \partial_t u + \partial_t w + \partial_x f(u) = 0, \] where $w(x,t) = [\mathcal{F}(u)(x,t)]$ is the output of a specific hysteresis operator, namely the Play…

We develop a fractional version of Hirota's bilinear calculus that is built directly from the spectral (Fourier-multiplier) fractional derivative on $\mathbb{R}$. For $0<\alpha\le 1$ we define \[ D_{\xi}^{\alpha}f\cdot g :=…

In this work, we will prove a uniqueness result for Calder\'on's inverse problem via some integral representation formulas for solutions of the Vekua equation in the framework of Clifford analysis.

We study the nodal deficiency of pairs of Neumann eigenfunctions defined over symmetric dumbbell domains. As the width of the connecting neck shrinks, these eigenfunctions converge to Neumann eigenfunctions defined over the ends of the…

This paper compares two similar diffusion equations that appear in meteorology. One is the quasi-geostrophic equation, and the other is the convection-diffusion equation. Both are two-dimensional bilinear equations, and the order of…

This paper investigates the dynamics of vegetation patterns in water-limited ecosystems using a generalized Klausmeier model that incorporates non-local plant dispersal within a finite habitat. We establish the well-posedness of the system…

We consider the wave propagation for a reaction-diffusion equation on the real line, with a random drift and Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) type nonlinear reaction. We show that when the average drift is positive, the…

We study observability for the one-dimensional wave equation on the torus from spacetime measurable observation sets. While the Geometric Control Condition (GCC) provides a sufficient criterion in many classical settings, it is no longer…

We apply the moving plane method in integral forms to classify the positive solutions of the critical Hartree equation on Heisenberg group \begin{equation}\label{0.1}…

We investigate the Cauchy problem for a semilinear spatio--temporal fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^\alpha u + (-\Delta)^{\mathsf{s}} u = |u|^p + t^{\sigma}\,\mathbf{w}(x), \quad (t,x) \in…

In this paper we construct a novel technique for eliminating and recovering the pressure for a fluid-structure interaction model. This pressure elimination methodology is valid for general bounded Lipschitz domains. The specific…

David and Semmes proved that if all CZOs (of suitable dimension) are bounded with respect to an Ahlfors regular measure, then the measure is uniformly rectifiable. We extend this theorem to the parabolic space and the first Heisenberg…

By providing optimal or nearly optimal integral estimates, we show that every positive, bounded or moderately growing, local weak solution to the critical $p$-Laplace equation in $\mathbb{R}^n$, with $n\geq 3$, and whose infimum over a ball…

We study incompressible Euler equations in $\mathbb{R}^d$ with $d \ge 4$ under bi-rotational symmetry without swirl, which reduces the Euler equations to a scalar vorticity advection in the first quadrant. We show that patch type initial…

The paper is concerned with a posteriori estimates for approximations of boundary value problems generated by the spectral fractional Laplace operator. The derivation is based upon the Stinga--Torrea extension, which generalizes the…