Global weak solutions with higher regularity to the compressible Navier-Stokes equations under Dirichlet boundary conditions

Abstract

In this manuscript, we aim to establish global existence of weak solutions with higher regularity to the compressible Navier-Stokes equations under no-slip boundary conditions. Though Lions\cite{L1} and Feireisl\cite{F1} have established global weak solutions with finite energy under Dirichelet boundary conditions by making use of so called effective viscous flux and oscillation defect measure,Hoff has investigated global weak solutions with higher regularity in \cite{H1,Hof2} when the domain is either whole space or half space with Navier-slip boundary conditions, yet the existence theory of global weak solution with higher regularity under Dirichlet boundary conditions remains unknown. In this paper we prove that the system will admit at least one global weak solutions with higher regularity as long as the initial energy is suitably small when the domain is a 2D solid disc. This is achieved by exploiting the structure of the exact Green function of the disc to decompose the effective viscous flux into three parts, which corresponds to the pressure term, boundary term and the remaining term respectively. In order to control the boundary term, one of the key observations is to use the geometry of the domain which sucessfully to bound the integral of the effective viscous flux where $L^1$ norm is always unbounded.