Blow Up Criterion For The Compressible Isentropic Fluid Dynamics Equations With Initial Vacuum

In this paper,we mainly study blow-up criterion for strong solutions to 3-dimensional compressible Navier-Stokes equations when the initial vacuum is allowed.We propose a blow-up criterion for strong solutions to the Cauchy problem of 3-dimensional isentropic compressible Navier-Stokes equations in terms of the integral norm of density p and the divergence of the velocity u.This paper refers to the research methods of Choe-Yang[28],and the main purpose is to improve the existing results in the field that blow-up criterion for strong solutions to the Cauchy problem of those equations.We mainly discuss the following isentropic compressible Navier Stokes equations on(0,T)x R3:(?)where ?=?(t,x),u=(u1,u2,u3)(t,x)5 p=p(t,x)denote the density,velocity and pressure of the fluid,respectively.In this paper,we consider the Cauchy problem with(?,u)vanishes at infinity.When the initial vacuum is allowed,if T*<oo is the maximal existence time of strong solutions,then we prove that:(?) the result shows that if sup (?) is finite,then there exists a unique strong solution of 3-dimensional viscous isentropic compressible Navier Stokes equations on(0,T)x R3.