One of the most intriguing issues in the mathematical theory of the stationary Navier–Stokes equations is the regularity of weak solutions. This problem has been deeply investigated for homogeneous fluids. In this paper, the regularity of the solutions in the case of not constant viscosity is analyzed. Precisely, it is proved that for a bounded domain $\Omega \subset {\mathbb{R}}^2$, a weak solution $u\in W^{1,q}(\Omega )$ is locally Hölder continuous if $q=2$, and Hölder continuous around x, if $q\in (1,2)$ and $|\mu \left(x\right)-{\mu }_0|$ is suitably small, with ${\mu }_0$ positive constant; an analogous result holds true for a bounded domain $\Omega \subset {\mathbb{R}}^n(n>2)$ and weak solutions in $W^{1,n/2}(\Omega )$.

中文翻译：

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非齐次平稳 Navier-Stokes 方程弱解的规律

平稳 Navier-Stokes 方程的数学理论中最有趣的问题之一是弱解的规律性。这个问题已经针对均质流体进行了深入研究。本文分析了非恒定粘度情况下溶液的规律性。准确地证明，对于有界域$\Omega \subset {\mathbb{电阻}}^2$，弱解 $你\in {宽}^{1,q}(\Omega )$ 是局部 Hölder 连续的，如果 $q=2$, 并且 Hölder 围绕x连续，如果$q\in (1,2)$ 和 $|\mu \left(X\right)-{\mu }_0|$ 适当地小，与 ${\mu }_0$正常数；类似的结果适用于有界域$\Omega \subset {\mathbb{电阻}}^n(n>2)$ 和弱解 ${宽}^{1,n/2}(\Omega )$.