Mathematical justification of the point vortex dynamics in background fields on surfaces as an Euler-Arnold flow

Abstract

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.