Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids

Abstract

International audienceno abstrac

Compressible Fluids Driven by Stochastic Forcing: The Relative Energy Inequality and Applications

“…Our main goal is to derive a relative energy inequality for system (1.1-1.3) analogous to that obtained in the deterministic case in [8]. For the sake of simplicity, we focus on the space-periodic boundary conditions yielding the physical space in the form of the "flat" torus…”

Dimension Reduction for Compressible Viscous Fluids

“…Motivated by [5] (see also Dafermos [1], Germain [6], Mellet, Vasseur [13]) we introduce the (scaled) relative entropy functional 6) along with the relative entropy inequality…”

“…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.…”