In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.

在本文中，我们研究了具有各向异性粘性张量的拟平稳可压缩Stokes方程弱解的整体存在性问题。关键思想是通过将与弱解序列相关的能量方程的极限与由弱解序列限制所验证的与系统相关的能量方程进行比较而获得的新身份。在弱解的稳定性的背景下，这使我们能够构建缺陷度量，该缺陷度量用于证明密度的紧凑性，因此使我们能够识别极限模型中的压力。通过这样做，我们避免使用所谓的有效通量。使用这个新工具 我们解决了一个开放性问题，即对于此类系统，Leray的解存在全局性，而对各向异性振幅没有任何限制。这提供了一种灵活而自然的方法来处理可压缩的拟线性斯托克斯系统，这些系统在生物学，多孔介质，超导电性或低雷诺数条件下的其他应用中非常重要。