Energy conservation for the weak solutions of the compressible Navier-Stokes equations

TL;DR

This paper establishes energy conservation for weak solutions of the compressible Navier-Stokes equations under broad conditions, including vacuum states and degenerate viscosities, advancing fundamental understanding in fluid dynamics.

Contribution

It provides the first proof of energy conservation for weak solutions of the compressible Navier-Stokes equations, applicable across different viscosities and dimensions.

Findings

Energy conservation holds for weak solutions with constant viscosities.

Energy may be conserved even on vacuum states.

Results are dimension-independent.

Abstract

In this paper, we prove the energy conservation for the weak solutions of the compressible Navier-Stokes equations for any time $t>0$, under certain conditions. The results hold for the renormalized solutions of the equations with constant viscosities, as well as the weak solutions of the equations with degenerate viscosity. Our conditions do not depend on the dimensions. The energy may conserve on the vacuum for the compressible Navier-Stokes equations with constant viscosities. Our results are the first ones on the energy conservation for the weak solutions of the compressible Navier-Stokes equations.