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

TL;DR

This paper proves the existence of global weak solutions with higher regularity for the compressible Navier-Stokes equations under Dirichlet boundary conditions in a 2D disc, using Green function techniques and domain geometry.

Contribution

It establishes the first existence result for higher regularity weak solutions under Dirichlet conditions in a bounded domain, specifically a 2D disc, by novel decomposition of the effective viscous flux.

Findings

Existence of higher regularity weak solutions in a 2D disc

Decomposition of viscous flux into pressure, boundary, and remaining parts

Use of domain geometry to control boundary terms

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.