Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

TL;DR

This paper investigates the well-posedness of the EPDiff equations derived from fractional Sobolev-type metrics on diffeomorphism groups, providing new mathematical insights and commutator estimates relevant to physics and shape analysis.

Contribution

It establishes local and global well-posedness results for EPDiff equations with pseudo-differential inertia operators, introducing new commutator estimates for elliptic operators.

Findings

Proves local well-posedness for certain metric orders

Establishes global well-posedness under specific conditions

Derives new commutator estimates for elliptic pseudo-differential operators

Abstract

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical physics and in the field of shape analysis and template registration. Depending on the order of the metric, we will prove both local and global well-posedness results for these equations. As a result of our analysis we will also obtain new commutator estimates for elliptic pseudo-differential operators.