We are concerned here with results of existence and regularity of solutions to a stationary MHD problem, with a possibly multi-connected 2D domain associated with Navier-type boundary conditions. The RHS f, corresponding to the external forces, is in ${\mathit{L}}^p\left(\Omega \right)$, with $p\ge 2$, without any restriction on its norm. Two main difficulties are encountered in solving these problems: the geometry of the domain and the non-linear term $(\mathit{u}\cdot \nabla )(\mathit{u}-\alpha \Delta \mathit{u})$. We prove the existence of a solution $(\mathit{u},\mathit{B})$ in ${\mathit{H}}^2\left(\Omega \right)\times {\mathbf{W}}^{2,r}\left(\Omega \right)$ for any finite $r$.

中文翻译：

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具有 Navier 类型边界条件的 MHD 流

我们在这里关注的是固定 MHD 问题的存在性和解的规律性的结果，其中可能具有与 Navier 型边界条件相关的多连接 2D 域。对应于外力的 RHS f位于${\mathit{大号}}^p\left(\Omega \right)$， 和$p\ge 2$, 对其范数没有任何限制。在解决这些问题时遇到了两个主要困难：域的几何和非线性项$(\mathit{你}\cdot \nabla )(\mathit{你}-\alpha \Delta \mathit{你})$. 我们证明存在解$(\mathit{你},\mathit{乙})$在${\mathit{H}}^2\left(\Omega \right)\times {\mathbf{W}}^{2,r}\left(\Omega \right)$对于任何有限$r$.