Dissipative solutions to the compressible isentropic Navier-Stokes equations

TL;DR

This paper proves the existence of dissipative solutions for the compressible isentropic Navier-Stokes equations using a compactness argument, extending the concept of dissipative solutions from incompressible to compressible flows.

Contribution

It introduces a method to establish dissipative solutions for the compressible Navier-Stokes equations via limit processes from approximate solutions.

Findings

Existence of dissipative solutions is proven.

Method extends the concept from incompressible to compressible flows.

Uses compactness arguments to pass to the limit.

Abstract

The existence of dissipative solutions to the compressible isentropic Navier-Stokes equations was established in this paper. This notion was inspired by the concept of dissipative solutions to the incompressible Euler equations of Lions (\cite{Lions-1996}, Section 4.4). Our method is to recover such solutions by passing to the limits from approximated solutions, thanks to compactness argument.