Semi-invariant Riemannian metrics in hydrodynamics

TL;DR

This paper investigates semi-invariant Riemannian metrics on the diffeomorphism group relevant to hydrodynamics, establishing well-posedness results and analyzing regularity properties for these metrics, which are less restrictive than fully invariant ones.

Contribution

It introduces and studies semi-invariant Sobolev metrics on the diffeomorphism group, providing new well-posedness results and insights into their regularity properties in hydrodynamic models.

Findings

Established local and some global well-posedness results for semi-invariant metrics.

Identified higher regularity requirements compared to fully invariant metrics.

Showed no regularity loss or gain along geodesics despite semi-invariance.

Abstract

Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa--Holm equations are well-studied examples.A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description.Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. There are, however, many hydrodynamic-type equations, notably shallow water models and compressible Euler equations, where the underlying infinite-dimensional Riemannian structure is not fully right invariant, but still semi-invariant with respect to the subgroup of volume preserving diffeomorphisms. Here we study such metrics. For semi-invariant metrics of Sobolev $H^k$-type we give local and some global well-posedness results for the geodesic initial value problem. We also give results in the presence of a potential functional (corresponding to the fluid's internal energy). Our study reveals many pitfalls in going from fully right invariant to semi-invariant Sobolev metrics; the regularity requirements, for example, are higher. Nevertheless the key results, such as no loss or gain in regularity along geodesics, can be adopted.