AI 中文总结
研究通过稳定流形方法求解非电阻磁流体动力学方程，证明对于定常欧拉方程的小正则解\(B\)，存在弛豫到\((0,B)\)的无穷维解族，给出全局正则解，还表明定常欧拉方程小正则解可通过MHD从一类磁场拓扑可达且磁力线拓扑在极限中保持。
AI 中文摘要
我们证明，对于定常欧拉方程的任何足够小且正则的解\(B\)，存在一族非电阻磁流体动力学方程（MHD）的无穷维解\((u,b)\)，它们弛豫到\((0,B)\)。更确切地说，当\(t \to +\infty\)时，\((u,b)\)以指数速度快速趋近于\((0,B)\)。这族解可视为位于非电阻MHD方程围绕平衡态\((0,B)\)的稳定流形上。该族解是否与稳定流形实际重合的问题仍然悬而未决。作为结果的副产品，我们给出了一大类非电阻MHD方程的全局正则解。另一个结果是，根据莫法特的定义，定常欧拉方程的任何足够小且正则的解都可以通过MHD从一大类磁场（非平凡地）拓扑可达，并且在这种情况下，磁力线的拓扑在\(t \to +\infty\)的极限中（完全）保持。
英文摘要
We prove that given any sufficiently small and regular solution $B$ of the stationary Euler equations there exists an infinite dimensional family of solutions $(u,b)$ of the non-resistive magnetohydrodynamics equations (MHD) that relax to $(0, B)$. More precisely, $(u,b) \to (0, B)$ exponentially fast as $t \to +\infty$. This family may be viewed as lying in the stable manifold of the non-resistive MHD equations around the equilibrium state $(0, B)$. The problem whether it actually coincides with the stable manifold remains open. As a byproduct of our result, we provide a large class of global regular solutions of the non-resistive MHD equations. Another consequence is that any sufficiently small and regular solution of the stationary Euler equation is (non-trivially) topologically accessible via MHD from a large class of magnetic fields according to the definition of Moffatt and, in this scenario, the topology of the magnetic lines is (entirely) preserved in the limit $t \to + \infty$.
CommentsThe order of Theorems 1.3 and 1.5 has been reversed. The former Theorem 1.3, now Theorem 1.4, has been strengthened. Minor revisions and typographical corrections have also been made
