Energy equalities for compressible Navier-Stokes equations

TL;DR

This paper establishes energy equalities for compressible Navier-Stokes equations with general pressure laws and degenerate viscosities, providing conditions on weak solution regularity that hold under various boundary conditions without boundary layer assumptions.

Contribution

It introduces a unified approach to prove energy equalities for these equations, applicable to different boundary conditions and avoiding boundary layer assumptions in bounded domains.

Findings

Energy equalities hold under specified regularity conditions.

Method applies to periodic and Dirichlet boundary conditions.

No boundary layer assumptions needed for bounded domains.

Abstract

The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity of weak solutions for these equalities to hold. The method of proof is suitable for the case of periodic as well as homogeneous Dirichlet boundary conditions. In particular, by a careful analysis using the homogeneous Dirichlet boundary condition, no boundary layer assumptions are required when dealing with bounded domains with boundary.