Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D

TL;DR

This paper proves the existence of global strong solutions for 1D compressible Navier-Stokes equations with degenerate viscosity, using a new formulation involving an effective velocity to control vacuum states.

Contribution

It introduces a novel formulation with an effective velocity that allows handling vacuum and large initial data in 1D compressible Navier-Stokes equations.

Findings

Existence of global strong solutions for large initial data

Effective velocity formulation controls vacuum states

Density satisfies a parabolic equation

Abstract

We consider Navier-Stokes equations for compressible viscous fluids in one dimension. We prove the existence of global strong solution with large initial data for the shallow water system. The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity $v$ (see \cite{cras,para,CPAM,CPAM1}) such that the density verifies a parabolic equation. We estimate $v$ in $L^\infty$ norm which enables us to control the vacuum on the density.