Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities

TL;DR

This paper proves the global existence of weak solutions for the barotropic compressible Navier-Stokes equations with degenerate viscosities, addressing an open problem in mathematical fluid mechanics.

Contribution

It establishes the existence of weak solutions with degenerate viscosities in 2D and 3D, using novel approximation and compactness techniques.

Findings

Constructed approximate systems with energy and entropy inequalities

Applied Mellet-Vasseur compactness results to degenerate viscosities

Proved global weak solutions for large initial data in periodic and whole space domains

Abstract

This paper concerns the existence of global weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients. We construct suitable approximate system which has smooth solutions satisfying the energy inequality, the BD entropy one, and the Mellet-Vasseur type estimate. Then, after adapting the compactness results due to Mellet-Vasseur [Comm. Partial Differential Equations 32 (2007)], we obtain the global existence of weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in two or three dimensional periodic domains or whole space for large initial data. This, in particular, solved an open problem in [P. L. Lions. Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford University Press, 1998].