Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations

Abstract

Abstract. In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation [2]. The main contribution of this paper is to derive the Mellet-Vasseur type inequality [32] for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible Navier-Stokes equations, for any γ > 1 in two dimensional space and for 1 < γ < 3 in three dimensional space, with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions in [27].

Existence of global strong solution for the compressible Navier–Stokes equations with degenerate viscosity coefficients in 1D

“…Let us mention in particular that the case with and verifies the algebraic relation related to the BD entropy discovered in , it corresponds here to the so called viscous shallow water system. For the important problem of the existence of global weak solutions has been recently resolved independently by Vasseur and Yu and Li and Xin in .…”

Global Weak Solutions to the Compressible Quantum Navier--Stokes Equations with Damping

“…Remark 1.1 It should be noted that the arguments in Vasseur-Yu [34,35] rely crucially on the assumption that the gradient of velocity field ∇u is a well-defined function, which indeed does not make sense in the presence of vacuum. In particular, in the proof of [35,Lemma 4.2], which is crucial to deduce the key Mellet-Vasseur type estimate in [35], it requires essentially that ∇u is a well-defined function. Very recently, Lacroix-Violet & Vasseur [25] also study the QNS equations and consider a new…”

“…In the absence of damping terms, we need further to derive the Mellet-Vasseur type estimate. As pointed in [4,5,35], the third order dispersive term prevents one from obtaining directly a Mellet-Vasseur type inequality. This difficulty is overcome by deriving the Mellet-Vasseur type estimate on (ρ, w) to the transformation system (2.67) without third order term.…”

Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications

“…Remark that existence of κ-entropic solution for the compressible Navier-Stokes equations with µ( ) = µ with µ constant and λ( ) = 0 without extra terms (capillary, drag, singular pressure) has been recently proved in the nice paper [22].…”

“…In particular, we are able to handle the case µ( ) = µ and a general pressure p( ) strictly monotone like p( ) = a γ with γ > 1. This corresponds to the compressible Navier Stokes system considered by A. Vasseur and C. Yu in [22]. By a combination of estimates based on the so called B-D (Bresch-Desjardin) entropy [4], Mellet-Vasseur estimates [17] and some original renormalization techniques, they recently obtained the first existence result of global weak solutions without additional regularizing/damping terms (friction, surface tension).…”