Global existence of weak solutions for the anisotropic compressible Stokes system

Abstract

In this paper, we prove global existence of weak solutions for the stationary compressible Navier-Stokes equations with an anisotropic and nonlocal viscous term in a periodic domain T 3 . This gives an answer to an open problem important for applications in geophysics or in microfluidics. The main idea is to adapt in a non-trivial way the new idea developped by the authors in a previous paper, see [2] which allowed them to treat the anisotropic compressible quasi-stationary Stokes system. 1 3 1 + √ 13 ≈ 1.53 for volume non-potential forces respectively for γ > 1 8 3 + √ 41 ≈ 1.175. Finally, the optimal result in the periodic framework, existence for γ > 1 was obtained in [15] by S. Jiang, and C. Zhou. Concerning finite domains with Dirichlet boundary condition, the optimal result regarding the value of γ is due to P. Plotnikov and W. Weigant [23] who constructed solutions for any f ∈ L ∞ (Ω), g = 0 with pressure functions p (ρ) = ρ γ for any γ > 1, improving upon previous preliminary results

Extension of the Hoff solutions framework to cover Navier-Stokes equations for a compressible fluid with anisotropic viscous-stress tensor

“…When dealing with weak solutions for non-linear PDE systems, one of the most delicate aspects is the stability analysis: given a sequence of weak solutions for some well-chosen approximated systems, show that this sequence converges to a solution for the initial system. The key ingredient in [BB20] and [BB21] is an identity that we found when comparing on the one hand, the limiting energy equation and on the other hand, the equation of the energy associated to the limit system. In order to justify such an identity, a crucial assumption seems to be the fact that the pressure is L 2 , an apriori estimate which is ensured by basic a-priori estimates in the case of the Stokes system or for the stationary Navier-Stokes system.…”

“…Moreover, the restriction for the adiabatic coefficient γ given by (1.10) excludes most of the physically realistic values : monoatomic gases 5/3, ideal diatomic gases 7/5, viscous shallow-water γ = 2. Let us also mention our results concerning global weak solutions à la Leray for the quasi-stationary compressible Stokes in [BB20] where an anisotropic diffusion −div(AD(u)) is considered with no smallness assumption on the anisotropic amplitude needed and for the stationary compressible Navier-Stokes equations in [BB21] with a viscous diffusion operator given by −Au (under some constraints) where A is composed by a classical constant viscous part plus an anisotropic contribution and a possible nonlocal contribution.…”

“…This makes it impossible to write the energy equation because, loosely speaking, the velocity cannot be used as a test function in a weak-formulation of (1.1). Thus, it seems hopeless to justify the limiting passage as in [BB20] and [BB21] in the most general setting of weak-solutions à la Leray. Obviously, one may ask if we can work in an intermediate regularity setting.…”

Weak solutions for the stationary anisotropic and nonlocal compressible Navier-Stokes system

“…Anisotropic diffusion is present for instance in geophysical flows, see [22], while nonlocal diffusion is considered when studying confined fluids or in microfluidics where fluids flows thought narrow vessels. In order to achieve this goal, one key ingredient is the identity that we proposed in [2] which allowed us to give a simple proof for the existence of global weak-solutions for the anisotropic quasi-stationary Stokes system (compressible Brinkman equations).…”

“…Of course, the more delicate part is to recover uniform estimates with respect to the regularization parameter and to show that the limiting object is a solution of the stationary Navier-Stokes system. The first key ingredient in the proof of stability is the new identity discovered by the authors in [2] in the context of the compressible Brinkman system. As it turns out, the L 2 -integrability of the velocity field obtained via the basic energy estimate is not enough, better integrability is needed in order to justify rigorously the aforementioned identity.…”