Existence of weak solutions for compressible Navier-Stokes equations with entropy transport

TL;DR

This paper proves the existence of weak solutions for the compressible Navier-Stokes equations with variable entropy, addressing a less studied case where entropy satisfies only a transport equation, expanding the theoretical understanding of such systems.

Contribution

It establishes existence results for weak solutions in three formulations for the compressible Navier-Stokes system with variable entropy, covering the optimal range of adiabatic coefficients.

Findings

Existence of weak solutions in three formulations.

Results valid for the optimal adiabatic coefficient range.

Addresses a less studied entropy transport case.

Abstract

We consider the compressible Navier-Stokes system with variable entropy. The pressure is a nonlinear function of the density and the entropy/potential temperature which, unlike in the Navier-Stokes-Fourier system, satisfies only the transport equation. We provide existence results within three alternative weak formulations of the corresponding classical problem. Our constructions hold for the optimal range of the adiabatic coefficients from the point of view of the nowadays existence theory.