Vanishing Distance Phenomena and the Geometric Approach to SQG

  • Martin BauerEmail author
  • Philipp Harms
  • Stephen C. Preston


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.



We would like to thank Martins Bruveris, Stefan Haller, Robert Jerrard, Cy Maor, Peter Michor, and Gerard Misiołek for helpful comments and discussions during the preparation of this manuscript.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of MathematicsAlbert Ludwig University of FreiburgFreiburgGermany
  3. 3.Department of MathematicsBrooklyn College and Graduate Center, City University of New YorkNew York CityUSA

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