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

Abstract

In this paper, we prove existence of weak solutions for the stationary compressible Navier-Stokes equations with an anisotropic and nonlocal viscous stress tensor in a periodic domain T 3. This gives an answer to an open problem important for applications in geophysics or in microfluidics. One of the key ingredients is the new identity discovered by the authors in [2] which was used to study the non-stationary anisotropic compressible Brinkman system.

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.…”

“…Let us recall that classical techniques due to P.L. Lions [Lio96] and E. Feireisl [Fei01] do not apply in this context, see the discussions from the introductions of [BJ18], [BB20], [BB21] for more details. Moreover, the work by the first author and P.E.…”

“…Jabin requires a relative large γ and, maybe more importantly, as it was explained above, it is not straightforward to extend it to heterogenous in space anisotropic tensors (the fact that E can depend also on the space variable). Here, it is crucial to extend our idea from [BB20] that we successfully implemented in order to construct global weak solutions à la Leray for the Stokes-Brinkman system in [BB20] and for the stationary NS system in [BB21]. In these two papers, we did not need however to impose any restriction on the size of the initial data or the forcing terms.…”