Existence of Global Weak Solutions for 3D Degenerate Compressible Navier-Stokes Equations

TL;DR

This paper proves the existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations with large initial data, using entropy methods and inequalities, solving a long-standing open problem.

Contribution

It introduces a novel derivation of the Mellet-Vasseur inequality for weak solutions, enabling proof of global existence for any b3>1 in 3D.

Findings

Existence of global weak solutions for all b3>1

Applicable to large initial data with vacuum regions

Utilizes Bresch-Desjardins entropy conservation

Abstract

In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation. The main contribution of this paper is to derive the Mellet-Vasseur type inequality for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible Navier-Stokes equations, for any $\gamma>1$, in three dimensional space, with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions.