论文标题
作为欧拉 - 阿诺德流的背景字段中的点涡流动力学的数学合理性
Mathematical justification of the point vortex dynamics in background fields on surfaces as an Euler-Arnold flow
论文作者
论文摘要
表面上背景字段中的点涡流动力学在de rham电流的意义上是欧拉 - 阿诺德流。我们通过常规的simular分解制定了Euler-Arnold方程的电流值解。对于解决方案，我们首先证明，如果涡度的奇异部分是通过$ n = 1的$ q_n（t）$的线性组合给出的，则是$ n = 1，\ ldots，n $，$ q_n（t）$是点涡流方程的解决方案。相反，我们接下来证明，如果$ q_n（t）$是$ n = 1，\ ldots，n $的点涡流方程的解决方案，则存在着一个常规的singular-arnold方程的当前值的解决方案，并具有常规的单词分解，以使涡流的单一旋转部分由lineare combinition $ quitions $ quions $ q nefta unceptions $ q。作为推论，我们将Bernoulli定律概括为流场是弯曲表面并考虑到点涡流的存在的情况。从应用程序的角度来看，数学上的理由是重要的，因为单位球上旋转矢量场中的点涡流动力学被改编为地球物理流的数学模型，以便考虑到Inviscid流对Coriolis force的生效。
The point vortex dynamics in background fields on surfaces is justified as an Euler-Arnold flow in the sense of de Rham currents. We formulate a current-valued solution of the Euler-Arnold equation with a regular-singular decomposition. For the solution, we first prove that, if the singular part of the vorticity is given by a linear combination of delta functions centered at $q_n(t)$ for $n=1,\ldots,N$, $q_n(t)$ is a solution of the point vortex equation. Conversely, we next prove that, if $q_n(t)$ is a solution of the point vortex equation for $n=1,\ldots,N$, there exists a current-valued solution of the Euler-Arnold equation with a regular-singular decomposition such that the singular part of the vorticity is given by a linear combination of delta functions centered at $q_n(t)$. As a corollary, we generalize the Bernoulli law to the case where the flow field is a curved surface and where the presence of point vortices is taken into account. From the viewpoint of the application, the mathematical justification is of a significance since the point vortex dynamics in the rotational vector field on the unit sphere is adapted as a mathematical model of geophysical flow in order to take effect of the Coriolis force on inviscid flows into consideration.
