Weak-Strong uniqueness for compressible Navier-Stokes system with degenerate viscosity coefficient and vacuum in one dimension

TL;DR

This paper establishes weak-strong uniqueness for the one-dimensional compressible Navier-Stokes equations with degenerate viscosity and vacuum, using relative entropy and a new effective velocity formulation.

Contribution

It introduces a novel approach combining relative entropy with a new effective velocity to handle vacuum and degeneracy in the system, ensuring uniqueness.

Findings

Proves weak-strong uniqueness under specified conditions.

Handles initial vacuum density in the analysis.

Develops a new formulation making the system parabolic and hyperbolic.

Abstract

We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficient and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in \cite{Jiu} so that it is unique. The novelty consists in dealing with initial density $\rho_0$ which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al and Mellet and Vasseur (see \cite{PG,Fei,15}) combined with a new formulation of the compressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introduce a new effective velocity which makes the system parabolic on the density and hyperbolic on this velocity).