Analysis of PDEs (math.AP)

Abstract: The study of flows over an obstacle is one of the fundamental problems in fluids. In this paper we establish the global validity of the diffusive limit for the Boltzmann equations to the Navier-Stokes-Fourier system in an exterior domain. To overcome the well-known difficulty of the lack of Poincare's inequality, we develop a new $L^2-L^6$ splitting for dissipative hydrodynamic part Pf for nonlinear closure.

Abstract: The paper concerns about the uniform decaying property (abbreviated by UDP) of solutions for an anisotropic viscoelastic system in the form of integrodifferential system (abbreviated by VID system) with mixed type boundary condition. The mixed type condition consists of the homogeneous displacement boundary condition and a homogeneous traction boundary condition or with a dissipation. By using a dissipative structure of this system, we will prove the UDP in a unified way for the two cases, which are, when the time derivative of relaxation tensor decays with polynomial order and it decays with exponential order.

Abstract: We consider Orlicz--Laplace equation $-div(\frac{\varphi'(|\nabla u|)}{|\nabla u|}\nabla u)=f$ where $\varphi$ is an Orlicz function and either $f=0$ or $f\in L^\infty$. We prove local second order regularity results for the weak solutions $u$ of the Orlicz--Laplace equation. More precisely, we show that if $\psi$ is another Orlicz function that is close to $\varphi$ in a suitable sense, then $\frac{\psi'(|\nabla u|)}{|\nabla u|}\nabla u\in W^{1,2}_{loc}$. This work contributes to the building up of quantitative second order Sobolev regularity for solutions of nonlinear equations.

Abstract: We study some basic properties of generalized Morrey spaces $\mathcal{M}^{p,\phi}(\R^{d})$. Also, the problem $-\mbox{div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u=0$ in $\Omega$, where $\Omega$ is a bounded open set in $\R^d$, and potential $V$ is assumed to be not equivalent to zero and lies in $\mathcal{M}^{p,\phi}(\Omega)$, is studied. Finally, we establish the strong unique continuation for the $p$-Laplace operator in the case $V\in\mathcal{M}^{p,\phi}(\R^d)$.