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

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.

Fractional Sobolev metrics on spaces of immersions

“…On Diff(M ) we obtain local wellposedness of the geodesic equation for Sobolev metrics of order p ∈ [1/2, ∞); see Corollary 5.1. Analogous results have been obtained by different methods (smoothness of right-trivializations) for inertia operators that are defined as abstract pseudo-differential operators [24,10,3]. • For M = S 1 , our result specializes to the space of immersed loops in N .…”

“…In Eulerian coordinates, this equation is called Euler-Arnold [2] or EPDiff [29] In Lagrangian coordinates, the equation takes the form shown in the following corollary. The conditions for local well-posedness in this corollary agree with the ones in [3], where metrics governed by a general class of pseudo-differential operators are investigated. The proof is an application of It remains to verify these conditions for the specific operator P = (1 + ∆) p .…”

“…In Lagrangian coordinates, the equation takes the form shown in the following corollary. The conditions for local well-posedness in this corollary agree with the ones in [3], where metrics governed by a general class of pseudo-differential operators are investigated. The proof is an application of ḡ(P ϕ ϕ t , ϕ t ) vol g dt if and only if it satisfies the geodesic equation…”

Fractional Sobolev metrics on spaces of immersed curves

“…for some constant C that is independent of the curve c. 5 This implies (5.1), and hence the boundedness of c → ℓ c by Lemma 5.9. The proof of (5.2) now follows in the same manner as the length weighted case.…”

“…Local well-posedness in this setup was established for a wide variety of invariant metrics, typically using an Ebin-Marsden type analysis [20,29,23,28,7]. The focus of this article is geodesic and metric completeness, which is well understood for strong enough metrics in the case of diffeomorphism groups [39,30,29,16,5], but is mostly open for spaces of immersions. For closed, regular curves with values in Euclidean space, a series of completeness results both on the space of parametrized and unparametrized curves has been obtained, beginning with Bruveris, Michor and Mumford [14], see also [12,15,8].…”