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Refined blow up criteria for the full compressible Navier-Stokes equations involving temperature

Quansen Jiu, Yanqing Wang, Yulin Ye

Abstract

In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier-Stokes equations, we improve almost all the blow up criteria involving temperature to allow the temperature in its scaling invariant space for the 3D full compressible Navier-Stokes equations. Enlightening regular criteria via pressure $Π=\frac{\text {divdiv}}{-Δ}(u_{i}u_{j})$ of the 3D incompressible Navier-Stokes equations on bounded domain, we generalize Beirao da Veiga's result in [1] from the incompressible Navier-Stokes equations to the isentropic compressible Navier-Stokes system in the case away from vacuum.

Refined blow up criteria for the full compressible Navier-Stokes equations involving temperature

Abstract

In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier-Stokes equations, we improve almost all the blow up criteria involving temperature to allow the temperature in its scaling invariant space for the 3D full compressible Navier-Stokes equations. Enlightening regular criteria via pressure of the 3D incompressible Navier-Stokes equations on bounded domain, we generalize Beirao da Veiga's result in [1] from the incompressible Navier-Stokes equations to the isentropic compressible Navier-Stokes system in the case away from vacuum.

Paper Structure

This paper contains 6 sections, 9 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Notations and some auxiliary lemmas
  3. Blow up criteria with vacuum
  4. Extra constraint on the coefficients of viscosity
  5. Without extra constraint on the coefficients of viscosity
  6. Blow up criteria without vacuum

Key Result

Theorem \oldthetheorem

Suppose $(\rho,u,\theta)$ is the unique strong solution in Theorem localwith vacuum and $\lambda<3\mu$. If the maximal existence time $T^*$ is finite, then there holds for some $p, q$ satisfying

Theorems & Definitions (22)

  • Theorem \oldthetheorem
  • Remark 1.1
  • Remark 1.2
  • Theorem \oldthetheorem
  • Remark 1.3
  • Theorem \oldthetheorem
  • Remark 1.4
  • Remark 1.5
  • Corollary \oldthetheorem
  • Remark 1.6
  • ...and 12 more