Navier-Stokes-Fourier system with Dirichlet boundary conditions

We consider the Navier-Stokes-Fourier system describing the motion of a compressible, viscous, and heat-conducting fluid in a bounded domain Omega subset of R-d, d = 2, 3, with general non-homogeneous Dirichlet boundary conditions for the velocity and the absolute temperature, with the associated boundary conditions for the density on the inflow part. We introduce a new concept of a weak solution based on the satisfaction of the entropy inequality together with a balance law for the ballistic energy. We show the weak-strong uniqueness principle as well as the existence of global-in-time solutions.