Local well-posedness of the EPDiff equation: a survey

TL;DR

This survey reviews recent advances in establishing local well-posedness for the EPDiff equation, focusing on geodesic existence on diffeomorphism groups with non-local Fourier multiplier inertia operators.

Contribution

It compiles recent results on local existence of geodesics for the EPDiff equation with non-local Fourier multipliers on diffeomorphism groups.

Findings

Local well-posedness established for EPDiff with non-local Fourier multipliers

Existence of geodesics on $ ext{Diff}( or^d)$ and $ ext{Diff}( e^d)$ confirmed

Survey consolidates recent progress in the field

Abstract

This article is a survey on the local well-posedness problem for the general EPDiff equation. The main contribution concerns recent results on local existence of the geodesics on $\mathrm{Diff}(\mathbb{T}^{d})$ and $\mathrm{Diff}(\mathbb{R}^{d})$ when the inertia operator is a non-local Fourier multiplier.