可压缩Navier-Stokes方程无滑移边值问题的高正则整体弱解

Global weak solutions with higher regularity to the compressible Navier-Stokes equations under the no-slip boundary conditions

本文建立无滑移边界条件下可压缩Navier-Stokes方程的高正则弱解的整体存在性.Lions和Feireisl分别通过引入有效粘性通量和振荡缺陷测度,在无滑移边界条件下建立了允许真空初值的有限能量的整体弱解,而Hoff研究了当定义域为全空间或半空间且具有Navier滑移边界条件时的具有更高正则性的整体弱解理论.然而在无滑移边界条件下,具有更高正则性的整体弱解的存在性理论仍然未知.本文首次证明了当区域为二维实心圆盘,初始密度允许真空且初始能量足够小时,带有无滑移边界条件的可压缩等熵Navier-Stokes方程至少存在一个高正则性的整体弱解.该弱解的正则性介于由Lions和Feireisl引入的有限能量的弱解和Hoff的密度有界的弱解之间.本文的主要想法是利用圆盘的精确Green函数结构,将有效粘性通量分解为压力项、边界项和冗余项.为了控制边界项,我们一个关键的观察是利用圆盘的几何结构来精确控制有效粘性通量在边界的积分.

In this paper,we establish the global existence of weak solutions with higher regularity to the compressible Navier-Stokes equations under the no-slip boundary conditions.Lions and Feireisl have established global weak solutions with finite energy under the Dirichlet boundary conditions by making use of effective viscous flux and oscillation defect measure,while Hoff has investigated global weak solutions with higher regularity when the domain is either whole space or half space with Navier-slip boundary conditions,yet the existence theory of global weak solutions with higher regularity under the Dirichlet boundary conditions remains unknown.In this paper,we prove that the system will admit at least one global weak solution 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,the 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 successfully bounds the integral of the effective viscous flux on the boundary.