Vanishing distance phenomena and the geometric approach to SQG

Abstract

In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics. These include in particular the modified Constantin-Lax-Majda and surface quasi-geostrophic equations. The main result of this article shows that both of these equations stem from a Riemannian metric with vanishing geodesic distance.

arXiv comment17 pages, 3 tables, 3 figures

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Cited by in corpus (10)

• Fractional Sobolev metrics on spaces of immersions
• Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups
• Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
• The $L^p$-Fisher-Rao metric and Amari-Cencov $\alpha$-connections
• Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics
• Regularity and completeness of half-Lie groups
• A diffeomorphism-invariant metric on the space of vector-valued one-forms
• Incompressible Euler equations with stochastic forcing: a geometric approach
• On the $H^1(ds)$-gradient flow for the length functional
• The Michor-Mumford conjecture in Hilbertian H-type groups