In this paper, we prove the existence of weak solutions for the stationary compressible Navier-Stokes equations with an anisotropic and nonlocal viscous stress tensor in a periodic domain ${\mathbb{T}}^3$. This gives an answer to an open problem which is important for applications for instance in geophysics or in microfluidics. When dealing with weak solutions for such non-linear PDE system, the most delicate aspect 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. One of the key ingredients in the proof of stability is the adaptation of a new identity discovered by the authors [2] which was developed to study the quasi-stationary anisotropic compressible Brinkman system. This identity is used in order to recover strong convergence properties for the sequence of velocities and recover a posteriori strong convergence for the sequence of densities.

在本文中，我们证明了周期域中具有各向异性和非局部粘性应力张量的平稳可压缩Navier-Stokes方程的弱解的存在 ${\mathbb{Ť}}^3$。这为一个悬而未决的问题提供了答案，这个问题对于例如地球物理学或微流体学中的应用很重要。在处理此类非线性PDE系统的弱解时，最微妙的方面是稳定性分析：给定一些精心选择的近似系统的弱解序列，表明该序列收敛到初始系统的解。稳定性证明中的关键要素之一是对作者发现的新身份的适应[2]，该新身份是为研究准平稳各向异性可压缩Brinkman系统而开发的。使用该同一性是为了恢复速度序列的强收敛性，并恢复密度序列的后验强收敛。