On the Cauchy problem of the full Navier-Stokes equations for three-dimensional compressible viscous heat-conducting flows subject to large external potential forces

Abstract

Abstract. In this paper, we consider the Cauchy problem of the full Navier-Stokes equations for three-dimensional compressible viscous heat-conducting flows subject to external potential forces in the whole space R3 . For discontinuous data with small energy and vacuum, the global “intermediate weak” solutions with large oscillations and large external potential forces are obtained, provided the unique steady state is strictly away from vacuum. Moreover, if ‖∇ρ0‖L2∩Lp with any p ∈ (3, 6), ‖∇u0‖L3 and ‖∇θ0‖L2 are bounded, then the weak solution becomes a strong one belonging to a class of functions in which the uniqueness can be shown to hold, when the density is strictly away from vacuum and the viscosity coefficients satisfy 7μ > λ additionally.