Blowup Criterion for the Compressible Flows with Vacuum States

TL;DR

This paper establishes a criterion based on the deformation tensor norm that predicts the breakdown of smooth solutions in 3D compressible Navier-Stokes equations, including cases with vacuum states, extending previous incompressible results.

Contribution

It introduces a blowup criterion for compressible flows with vacuum, generalizes previous results, and allows initial vacuum states, advancing understanding of solution regularity.

Findings

Deformation tensor norm controls solution breakdown.

Criterion applies to flows with vacuum states.

Method extends to full compressible Navier-Stokes system.

Abstract

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional compressible Navier-Stokes equations, which will happen, for example, if the initial density is compactly supported \cite{X1}. More precisely, if a solution of the compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce's criterion for 3-dimensional incompressible Euler equations (\cite{po}). Moreover, our method can be generalized to the full Compressible Navier-Stokes system which improve the previous results. In addition, initial vacuum states are allowed in our cases.