Relative Entropies, Suitable Weak Solutions, and Weak-Strong Uniqueness for the Compressible Navier–Stokes System

Abstract

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.

Robustness of strong solutions to the compressible Navier-Stokes system

“…Note that, in accordance with the general weak-strong uniqueness property established in [8], the weak and strong solution emanating from the same initial data coincide as long as the latter exists.…”

Dimension Reduction for Compressible Viscous Fluids

“…In view of the above mentioned difficulties related to the validity of (1.8) or (1.9), our approach relies on the structural stability of the family of solutions of the barotropic Navier-Stokes system encoded in the relative entropy inequality introduced in [4], [5]. This method is basically independent of the specific form of the viscous stress and of possible "dissipative" bounds for the Navier-Stokes system.…”

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

“…Indeed we show that the dissipative weak solutions, meaning the weak solutions satisfying (1.17), enjoy the weak-strong uniqueness property -they coincide with the strong solution emanating from the same initial data as long as the latter exists. This result will be a direct consequence of the method of relative entropies adapted from [13], [14]. Finally, we note that even the dissipative weak solution may fail to be unique, at least for certain (non-smooth) initial data.…”

“…is a bounded energy weak solution to the Euler-Korteweg-Poisson system (1.1-1.3), (1.6) if 13) and the following integral identities…”