In this paper we study the global existence and uniqueness of classical solutions to the Cauchy problem for 3D isentropic compressible Navier-Stokes equations with general initial data which could contain vacuum.We give the relation between the viscosity coefficients and the initial energy,which implies that the Cauchy problem under consideration has a global classical solution.
Abstract In this paper, we study the stability of solutions of the Cauchy problem for 1-D compressible Narvier- Stokes equations with general initial data. The asymptotic limit of solution is found, under some conditions. The results in this paper imply the case that the limit function of solution as t → ee is a viscous contact wave in the sense, which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero. As a by-product, the decay rates of the solution for the fast diffusion equations are also obtained. The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.
In this paper,we prove a blow-up criterion of strong solutions to the 3-D viscous and non-resistive magnetohydrodynamic equations for compressible heat-conducting flows with initial vacuum.This blow-up criterion depends only on the gradient of velocity and the temperature,which is similar to the one for compressible Navier-Stokes equations.
