Study On Behavior Of Solutions Of Compressible Navier-Stokes Equations With Weak Viscosity And Heat Conductivity

It is well known that the full compressible Navier-Stokes equations with viscosity and heat conductivity coefficients of order of the Knudsen number > 0 can be deduced from the Boltzmann equation via the Chapman-Enskog expansion.In this paper,we carry out the rigorous mathematical study for initial-boundary value problems of the compressible Navier-Stokes equations.We construct the existence and most importantly obtain the higher regularities of the solutions of the full compressible Navier-Stokes system with weak viscosity and heat conductivity in a general bounded domain.The full compressible Navier-Stokes equations we studied in this paper have weak viscosity and weak conductivity.Standard ellipse estimation cannot be used directly,and ordinary regularity estimation cannot be used to deal with the boundary value problem of elliptic partial differential equations.To overcome these problems,we introduce the Helmholtz decomposition,Galerkin method,and the conormal derivatives to process the boundary terms and obtain higher-order energy estimates.