Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity

Abstract

We study the global existence and convergence rates of solutions to the three-dimensional compressible Navier-Stokes equations without heat conductivity, which is a hyperbolic-parabolic system. The pressure and velocity are dissipative because of the viscosity, whereas the entropy is non-dissipative due to the absence of heat conductivity. The global solutions are obtained by combining the local existence and a priori estimates if H 3 -norm of the initial perturbation around a constant state is small enough and its L 1 -norm is bounded. A priori decay-in-time estimates on the pressure and velocity are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.