Groups of Asymptotic Diffeomorphisms

TL;DR

This paper introduces classes of asymptotic diffeomorphisms with weighted Sobolev remainders, forming topological groups suitable for fluid dynamics applications like the Camassa-Holm and Euler equations.

Contribution

It establishes that these classes form topological groups under composition, enabling their use in geometric fluid dynamics models.

Findings

Classes form topological groups under composition

Applicable to fluid dynamics equations

Supports analysis of asymptotic behaviors at infinity

Abstract

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the method of V. Arnold. Specific applications have been obtained for the Camassa-Holm equation and the Euler equations.