Global Classical Solution To The Compressible Navier-Stokes Equations With Large Initial Data And Vacuum

This thesis is concerned with the compressible isentropic Navier-Stokes equations. The Navier-Stokes equations are the basic model describing the motion of viscous fluid substances. The Navier-Stokes equations are not only of the scientific and engineering interest but also of great interest in a purely mathematical sense. The research on the Navier-Stokes equations is one of the most activite research hotspots in mathematics. And there are many challenging open problems in this direction. In this paper, we investigate the existence of a global classical solution to 3D Cauchy problem of the isentropic compressible Navier-Stokes equations with large initial data and vacuum. There are two parts in this paper.In Chapter 2, we consider the exsitence of classical solutions to the isentropic compressible Navier-Stokes equations, when the far-field density is vacuum (ρ=0). In particular, we get the existence of global classical solutions under the assumption that (γ-1)1/3E0μ-1 is suitably small.In Chapter 3, we consider the classical solution to the Cauchy problem on the isentropic compressible Navier-Stokes equations in the case that the far-field density is away from vacuum (p>0). And specifically, we get the existence of global classical solutions, when (γ-1)1/36+ρ1/6)E01/4μ-1/3 is suitably small.The above results show that the initial energy E0 can be large if the adiabatic exponent γ is near 1 and the far-field density p is suitably small or the viscosity coefficient μ, is taken to be large. These results improve the one obtained by Huang-Li-Xin in [25], where the existence of the classical solution is proved with small initial energy. It should be pointed out that in the theorems obtained in this paper, no smallness restriction is put upon the initial data. The key part of the paper is to establish the time-independent upper bound of the density. And once that is obtained, the proof of the time-dependent higher norm estimatites of the desity and velocity follows in a similar way as in [25].