Two dimensional potential theory with a view towards vortex motion: Energy, capacity and Green functions

TL;DR

This paper explores two-dimensional potential theory's application to vortex motion, examining energy, capacity, and Green functions, and extending classical concepts to Riemann surfaces and the Schottky double.

Contribution

It introduces a unified approach connecting potential theory, Green functions, and vortex dynamics, including extensions to Riemann surfaces and the Schottky double.

Findings

Derived laws from Bernoulli's equation for vortex energy and forces.

Analyzed the role of Green functions in vortex motion and potential theory.

Extended classical potential theory concepts to Riemann surfaces and the Schottky double.

Abstract

The paper reviews some parts of classical potential theory with applications to two dimensional fluid dynamics, in particular vortex motion. Energy and forces within a system of point vortices are similar to those for point charges when the vortices are kept fixed, but the dynamics is different in the case of free vortices. Starting from Bernoulli's equation we derive these laws. Letting the number of vortices tend to infinity leads in the limit to considerations of capacity, harmonic measure and many other notions in potential theory. In particular various kinds of Green functions have a central role in the paper, where we make a difference between electrostatic and hydrodynamic Green function. We also consider the corresponding concepts in the case of closed Riemann surfaces provided with a metric. From a canonically defined monopole Green function we rederive much of the classical theory of harmonic and analytic forms. In the final section of the paper we return to the planar case, then reappearing in form of a symmetric Riemann surface, the Schottky double. Bergman kernels, electrostatic and hydrodynamic, come up naturally, and associated to the Green function the is a certain Robin function which is important for vortex motion and which also relates to capacity functions in classical potential theory.