Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

Abstract

We prove global existence of appropriate weak solutions for the compressible Navier-Stokes equations for more general stress tensor than those covered by P.-L. Lions and E. Feireisl's theory. More precisely we focus on more general pressure laws which are not thermodynamically stable; we are also able to handle some anisotropy in the viscous stress tensor. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).

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

“…In , Hoff extends the previous results by proving the existence of global weak solution with large discontinuous initial data having different limits at . In passing let us mention that the existence of global weak solution in any dimension has been proved for the first time by Lions in and the result has been later refined by Feireisl et al ( and , see also ). Concerning the uniqueness of the solution, Solonnikov in obtained the existence of strong solution for smooth initial data in finite time.…”

Global existence of weak solutions for the anisotropic compressible Stokes system

“…The smallness condition on the "amount" of anisotropy we can have in the system comes at this level. We point out that a similar condition on the anisotropy is imposed in [3] in order to treat the non-stationary compressible Navier-Stokes system.…”

“…There are two ways one can think of the anisotropic-effective flux. First, as explained in [3], we just take the divergence of the momentum equation and to write it as…”

Global Existence of Finite Energy Weak Solutions of Quantum Navier–Stokes Equations

“…Moreover, for some different choices of viscosity and capillarity coefficients the global existence has been proved in [9,8,14]. In general, the analysis of fluids with density dependent viscosity, even without capillary tensor, requires new tools compared to the case of constant viscosity, see Lions-Feireisl theory [31,19] and also [12]. This is due to a loss of control on uniform bounds on the velocity field in vacuum regions.…”