Geodesic completeness of the H^3/2 metric on Diff(S¹)

2019/04/29 by Martin Bauer, Bauer, Martin, Boris Kolev +5
Mathematics · #Analytic and geometric function theory #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1904.12523

arxiv created 2020/03/20 · arxiv updated 2020/03/23

Abstract

Of concern is the study of the long-time existence of solutions to the Euler--Arnold equation of the right-invariant H^3/2-metric on the diffeomorphism group of the circle. In previous work by Escher and Kolev it has been shown that this equation admits long-time solutions if the order s of the metric is greater than 3/2, the behaviour for the critical Sobolev index s=3/2 has been left open. In this article we fill this gap by proving the analogous result also for the boundary case. The behaviour of the H^3/2-metric is, however, still different from its higher order counter parts, as it does not induce a complete Riemannian metric on any group of Sobolev diffeomorphisms.

Citations

• Wave breaking for nonlinear nonlocal shallow water equations

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