The geometry of a vorticity model equation

TL;DR

This paper establishes a geometric interpretation of a vortex model equation as a geodesic flow on a diffeomorphism group, proving local existence of solutions within a Sobolev space framework.

Contribution

It rigorously links the modified Constantin-Lax-Majda equation to geodesic flows on diffeomorphism groups with a specific Sobolev metric, and proves local existence of solutions.

Findings

The equation models geodesic flow on a subgroup of diffeomorphisms.

Local existence of geodesics in Sobolev class H^k for k ≥ 2.

Connection between vortex dynamics and geometric structures.

Abstract

We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics describes the geodesic flow on the subgroup of orientation-preserving diffeomorphisms fixing one point, with respect to right-invariant metric induced by the homogeneous Sobolev norm $H^{1/2}$ and show the local existence of the geodesics in the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$.