Blow up criteria for the compressible Navier--Stokes equations

TL;DR

This paper introduces new criteria for predicting blow-up in solutions to the 3-D compressible Navier--Stokes equations, linking the integrability of density and velocity divergence to solution regularity.

Contribution

It proposes a novel blow-up criterion based on the integral norms of density and divergence, applicable without restrictions on viscosity constants.

Findings

New blow-up criterion for barotropic gases

Applicable to heat conducting gases

Links higher integrability to regularity

Abstract

We study the strong solution to the 3-D compressible Navier--Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density $\rho$ and the divergence of the velocity $\bu$ without any restriction on the physical viscosity constants. Our blow up criteria can be seen as a partial realization of the underlying principle that the higher integrability implies the boundedness and then eventual regularity. We also present similar blow up criterion for the heat conducting gases.