标题： 关于几何流体力学和无限维磁系统 显示英文标题

标题： On geometric hydrodynamics and infinite dimensional magnetic systems

摘要： 本文中，我们将 V. Arnold 的著名方法——通过描述赋右不变度量的李群上测地线流的欧拉–阿诺尔方程~\cite{Arnold66} ——与他在磁场中带电粒子运动的表述~\cite{ar61} 相结合。我们引入了~\emph{磁性欧拉-阿诺德方程}，这是赋右不变度量以及作为磁场的右不变闭二形式的李群上的无限维磁性测地线流的欧拉形式。作为说明，我们证明了 Korteweg–de Vries 方程、广义 Camassa–Holm 方程、无穷电导率方程和全局准地转方程都可以被解释为磁性欧拉–阿诺尔方程。特别是，我们得到了与全局准地转方程相关的磁性欧拉–阿诺尔方程的局部和整体适定性结果。 显示英文摘要

摘要： In this article, we combine V. Arnold's celebrated approach via the Euler--Arnold equation -- describing the geodesic flow on a Lie group equipped with a right-invariant metric~\cite{Arnold66} -- with his formulation of the motion of a charged particle in a magnetic field~\cite{ar61}. We introduce the \emph{magnetic Euler--Arnold equation}, which is the Eulerian form of the magnetic geodesic flow for an infinite-dimensional magnetic system on a Lie group endowed with a right-invariant metric and a right-invariant closed two-form serving as the magnetic field. As an illustration, we demonstrate that the Korteweg--de Vries equation, the generalized Camassa--Holm equation, the infinite conductivity equation, and the global quasi-geostrophic equations can all be interpreted as magnetic Euler--Arnold equations. In particular, we obtain both local and global well-posedness results for the magnetic Euler--Arnold equation associated with the global quasi-geostrophic equations.