Serrin-Type Criterion for the Three-Dimensional Viscous Compressible Flows

Abstract

We extend the well-known Serrin's blowup criterion for the three-dimensional (3D) incompressible Navier-Stokes equations to the 3D viscous compressible cases. It is shown that for the Cauchy problem of the 3D compressible Navier-Stokes system in the whole space, the strong or smooth solution exists globally if the velocity satisfies the Serrin's condition and either the supernorm of the density or the L 1 (0, T ; L ∞ )-norm of the divergence of the velocity is bounded. Furthermore, in the case that either the shear viscosity coefficient is suitably large or there is no vacuum, the Serrin's condition on the velocity can be removed in this criteria.

Blow-up criterion for compressible MHD equations

“…Recently, Huang and Xin [19] improved the results as The purpose of this paper is to derive the corresponding blow-up criteria for the compressible MHD equations in the whole space. Moreover, in this paper we shall study the blow-up criterion when the initial density vacuum is occured.…”

A Blow-up Criterion of Strong Solutions to the Compressible Viscous Heat-Conductive Flows with Zero Heat Conductivity

“…The blow-up criterion has been studied for the isentropic Navier-Stokes equations, see [9,[11][12][13]23].…”

Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum

“…So we only need to prove (4.4). As that in [19], the key point here is to estimate ∇u L 1 (0,T ;L ∞ ) and ∇ρ L ∞ (0,T ;L p ) with p ∈ [2, 6], which will be achieved by using the Beale-Kato-Majda's type inequality developed in [10,11].…”

“…(4.9) By the standard L p -estimate of elliptic system, we infer from (1.2) that ∇ 2 u L p C ρu L p + ∇ P L p + H∇ H L p . (4.10) In order to deal with ∇u L ∞ , we recall the following Beale-Kato-Majda's type inequality (see [10,11]):…”