Relative entropies, suitable weak solutions, and weak strong uniqueness for the compressible Navier-Stokes system

We introduce the notion of relative entropy for the weak solutions to the compressible Navier–Stokes system. In particular, we show that any finite energy weak solution satisfies a relative entropy inequality with respect to any couple of smooth functions satisfying relevant boundary conditions. As a corollary, we establish the weak-strong uniqueness property in the class of finite energy weak solutions, extending thus the classical result of Prodi and Serrin to the class of compressible fluid flows.