Vanishing distance phenomena and the geometric approach to SQG

TL;DR

This paper investigates the geometric properties of fractional Sobolev metrics on diffeomorphism groups, revealing that key fluid dynamics equations originate from metrics with vanishing geodesic distance, impacting the understanding of these PDEs.

Contribution

It demonstrates that certain fluid dynamics equations are derived from Riemannian metrics with vanishing geodesic distance, providing new geometric insights into these PDEs.

Findings

Modified Constantin-Lax-Majda equation arises from vanishing distance metric.

Surface quasi-geostrophic equation stems from a vanishing distance geometry.

Geometric interpretation links PDEs to fractional Sobolev metrics.

Abstract

In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics. These include in particular the modified Constantin-Lax-Majda and surface quasi-geostrophic equations. The main result of this article shows that both of these equations stem from a Riemannian metric with vanishing geodesic distance.