Global well-posedness of non-heat conductive compressible Navier-Stokes equations in 1D

Abstract

In this paper, the initial-boundary value problem of the 1D full compressible Navier-Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any $H^1$ initial data. The initial density is required to be nonnegative, which is not necessary to be uniformly away from vacuum. This not only generalizes the well-known result of Kazhikhov--Shelukhin (Kazhikhov, A.~V.; Shelukhin, V.~V.: \emph{Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas}, J.\,Appl.\,Math.\,Mech., \bf41 \rm(1977), 273--282.) from the heat conductive case to the non-heat conductive case, and the initial vacuum is allowed.