Mathematical theory of compressible, viscous, and heat conducting fluids

We review some recent results on the Navier–Stokes–Fourier system governing the evolution of a general compressible, viscous, and heat conducting fluid. We discuss several concepts of weak solutions, in particular, using the implications of the Second law of thermodynamics. We introduce the concept of relative entropy and dissipative solution and show the principle of weak-strong uniqueness. We relate these results to their counterparts of the inviscid (Euler) system. Other applications of the approach based on relative entropy concept will be given. In particular, we discuss the problems of model reduction and the related singular limits. Special attention is paid to singular limits towards the inviscid system.