Shinbrot Type Criteria for Energy Conservation of the Compressible Navier-Stokes Equations

TL;DR

This paper establishes a new, weaker regularity criterion ensuring energy conservation for weak solutions of the compressible Navier-Stokes equations, applicable to fluids with various viscosity properties.

Contribution

It introduces a novel Shinbrot-type regularity criterion that is less restrictive than previous conditions, broadening the class of solutions satisfying energy equality.

Findings

Proves energy equality under the new criterion for compressible fluids.

Applicable to fluids with both constant and degenerate viscosity.

Uses a novel weak-type commutator estimate.

Abstract

We prove that weak solutions to the compressible Navier-Stokes equations satisfy the energy equality under a Shinbrot-type regularity criterion. Our method applies to the fluids with both constant and degenerate viscosity and relies on a novel weak-type commutator estimate. These criterion are strictly weaker than those required in prior works [Arch. Ration. Mech. Anal., 225 (2017)] and [SIAM J. Math. Anal. 52 (2020)].