Global strong solutions to the 3D full compressible Navier–Stokes equations with density–temperature–dependent viscosities in bounded domains

Abstract

We consider the three–dimensional full compressible Navier–Stokes system with density–temperature–dependent viscosities in smooth bounded domains. For the case when the velocity u and absolute temperature θ admit the Dirichlet boundary condition, the strong solutions exist globally in time provided that ‖∇u0‖L22+‖∇θ0‖L22 is suitably small. Through some time–weighted a priori estimates, the main difficulties caused by the density–temperature–dependent viscosities and the bounded domain are overcome. Moreover, the time–uniform upper bounds for the Lp–norm of the gradient of the density are obtained, which is of independent interest for compressible fluids when initial vacuum is allowed.