Skip to main content
SpringerLink
Log in

Vanishing Distance Phenomena and the Geometric Approach to SQG

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. That is, the injectivity radius of \((M,\langle \cdot ,\cdot \rangle )\) is positive and each iterated covariant derivative of the curvature is uniformly bounded in the metric; see [22, 45] for more details. This is automatically the case if M is compact or Euclidean.

References

  1. Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble)16(fasc.1), 319–361, 1966

    MathSciNet  MATH  Google Scholar 

  2. Banyaga, A.: The Structure of Classical Diffeomorphism Groups, Mathematics and its Applications, vol. 400. Kluwer Academic Publishers Group, Dordrecht 1997

    MATH  Google Scholar 

  3. Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Glob. Anal. Geom. 41(4), 461–472, 2012

    MathSciNet  MATH  Google Scholar 

  4. Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. Ann. Glob. Anal. Geom. 44(1), 5–21, 2013

    MathSciNet  MATH  Google Scholar 

  5. Bauer, M., Bruveris, M., Michor, P.W.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II. Ann. Glob. Anal. Geom. 44(4), 361–368, 2013

    MathSciNet  MATH  Google Scholar 

  6. Bauer, M., Bruveris, M., Michor, P.W.: Homogeneous Sobolev metric of order one on diffeomorphism groups on real line. J. Nonlinear Sci. 24(5), 769–808, 2014

    ADS  MathSciNet  MATH  Google Scholar 

  7. Bauer, M., Escher, J., Kolev, B.: Local and global well-posedness of the fractional order EPDiff equation. J. Differ. Equ. 258(6), 2010–2053, 2015

    ADS  MathSciNet  MATH  Google Scholar 

  8. Bauer, M., Kolev, B., Preston, S.C.: Geometric investigations of a vorticity model equation. J. Differ. Equ. 260(1), 478–516, 2016

    ADS  MathSciNet  MATH  Google Scholar 

  9. Bruveris, M., Vialard, F.-X.: On completeness of groups of diffeomorphisms. J. Eur. Math. Soc. (JEMS)19(5), 1507–1544, 2017

    MathSciNet  MATH  Google Scholar 

  10. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664, 1993

    ADS  MathSciNet  MATH  Google Scholar 

  11. Castro, A., Córdoba, D.: Infinite energy solutions of the surface quasi-geostrophic equation. Adv. Math. 225(4), 1820–1829, 2010

    MathSciNet  MATH  Google Scholar 

  12. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243, 1998

    MathSciNet  MATH  Google Scholar 

  13. Constantin, A., Kolev, B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A35(32), R51–R79, 2002

    ADS  MathSciNet  MATH  Google Scholar 

  14. Constantin, P., Lax, P.D., Majda, A.: A simple one-dimensional model for the three-dimensional vorticity equation. Commun. Pure Appl. Math. 38(6), 715–724, 1985

    ADS  MathSciNet  MATH  Google Scholar 

  15. Ebin, D.G., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 2(92), 102–163, 1970

    MathSciNet  MATH  Google Scholar 

  16. Ebin, D.G., Misiołek, G., Preston, S.C.: Singularities of the exponential map on the volume-preserving diffeomorphism group. Geom. Funct. Anal. 16(4), 850–868, 2006

    MathSciNet  MATH  Google Scholar 

  17. Eichhorn, J.: Global Analysis on Open Manifolds. Nova Science Publishers Inc., New York 2007

    MATH  Google Scholar 

  18. Eliashberg, Y., Polterovich, L.: Bi-invariant metrics on the group of Hamiltonian diffeomorphisms. Int. J. Math. 4(5), 727–738, 1993

    MathSciNet  MATH  Google Scholar 

  19. Escher, J., Kolev, B.: Geodesic completeness for Sobolev \(H^s\)-metrics on the diffeomorphism group of the circle. J. Evol. Equ. 14(4–5), 949–968, 2014

    MathSciNet  MATH  Google Scholar 

  20. Escher, J., Kolev, B.: Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. J. Geom. Mech. 6(3), 335–372, 2014

    MathSciNet  MATH  Google Scholar 

  21. Escher, J., Kolev, B., Wunsch, M.: The geometry of a vorticity model equation. Commun. Pure Appl. Anal. 11(4), 1407–1419, 2012

    MathSciNet  MATH  Google Scholar 

  22. Greene, R.E.: Complete metrics of bounded curvature on noncompact manifolds. Arch. Math. 31(1), 89–95, 1978

    MathSciNet  MATH  Google Scholar 

  23. Hofer, H.: On the topological properties of symplectic maps. Proc. R. Soc. Edinb. Sect. A Math. 115(1–2), 25–38, 1990

    MathSciNet  MATH  Google Scholar 

  24. Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51(6), 1498–1521, 1991

    MathSciNet  MATH  Google Scholar 

  25. Inci, H., Kappeler, T., Topalov, P.: On the regularity of the composition of diffeomorphisms. Mem. Am. Math. Soc. 226(1062), vi+60, 2013

    MathSciNet  MATH  Google Scholar 

  26. Jerrard, R.L., Maor, C.: Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups. Ann. Glob. Anal. Geom. 56(2), 351–360, 2019

    MathSciNet  MATH  Google Scholar 

  27. Jerrard, R.L., Maor, C.: Geodesic distance for right-invariant metrics on diffeomorphism groups—critical Sobolev exponents. Ann. Glob. Anal. Geom. 55(4), 631–656, 2019

    MathSciNet  MATH  Google Scholar 

  28. Khesin, B., Lenells, J., Misiołek, G., Preston, S.C.: Curvatures of Sobolev metrics on diffeomorphism groups. Pure Appl. Math. Q. 9(2), 291–332, 2013

    MathSciNet  MATH  Google Scholar 

  29. Khesin, B., Misiołek, G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176(1), 116–144, 2003

    MathSciNet  MATH  Google Scholar 

  30. Khesin, B., Ovsienko, V.: The super Korteweg–de Vries equation as an Euler equation. Funktsional. Anal. i Prilozhen. 21(4), 81–82, 1987

    MathSciNet  MATH  Google Scholar 

  31. Kolev, B.: Local well-posedness of the EPDiff equation: a survey. J. Geom. Mech. 9(2), 167–189, 2017

    MathSciNet  MATH  Google Scholar 

  32. Kouranbaeva, S.: The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40(2), 857–868, 1999

    ADS  MathSciNet  MATH  Google Scholar 

  33. Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence 1997

    MATH  Google Scholar 

  34. Lang, S.: Real and Functional Analysis, Graduate Texts in Mathematics, vol. 142. Springer, New York 1993

    Google Scholar 

  35. Lang, S.: Fundamentals of Differential Geometry, Graduate Texts in Mathematics, vol. 191. Springer, New York 1999

    Google Scholar 

  36. Lee, J.M.: Geometric approach on the global conservative solutions of the Camassa–Holm equation. J. Geom. Phys. 142, 137–150, 2019

    ADS  MathSciNet  MATH  Google Scholar 

  37. Lenells, J.: The Hunter–Saxton equation describes the geodesic flow on a sphere. J. Geom. Phys. 57(10), 2049–2064, 2007

    ADS  MathSciNet  MATH  Google Scholar 

  38. Lenells, J.: Weak geodesic flow and global solutions of the Hunter–Saxton equation. Discrete Contin. Dyn. Syst. 18(4), 643–656, 2007

    MathSciNet  MATH  Google Scholar 

  39. Lenells, J.: The Hunter–Saxton equation: a geometric approach. SIAM J. Math. Anal. 40(1), 266–277, 2008

    MathSciNet  MATH  Google Scholar 

  40. McKean, H.P.: Breakdown of a shallow water equation. Asian J. Math. 2(4), 867–874, 1998. (Mikio Sato: a great Japanese mathematician of the twentieth century)

    MathSciNet  MATH  Google Scholar 

  41. McKean, H.P.: Breakdown of the Camassa–Holm equation. Commun. Pure Appl. Math. 57(3), 416–418, 2004

    MathSciNet  MATH  Google Scholar 

  42. Michor, P.W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245, 2005. (electronic)

    MathSciNet  MATH  Google Scholar 

  43. Misiołek, G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24(3), 203–208, 1998

    ADS  MathSciNet  MATH  Google Scholar 

  44. Misiołek, G., Preston, S.C.: Fredholm properties of Riemannian exponential maps on diffeomorphism groups. Invent. Math. 179(1), 191–227, 2010

    ADS  MathSciNet  MATH  Google Scholar 

  45. Müller, O., Nardmann, M.: Every conformal class contains a metric of bounded geometry. Math. Ann. 363(1–2), 143–174, 2015

    MathSciNet  MATH  Google Scholar 

  46. Preston, S.C., Washabaugh, P.: Euler–Arnold equations and Teichmüller theory. Differ. Geom. Appl. 59, 1–11, 2018

    MATH  Google Scholar 

  47. Ratiu, T., Schmid, R.: The differentiable structure of three remarkable diffeomorphism groups. Math. Z. 177(1), 81–100, 1981

    MathSciNet  MATH  Google Scholar 

  48. Shelukhin, E.: The Hofer norm of a contactomorphism. J. Symplectic Geom. 15(4), 1173–1208, 2017

    MathSciNet  MATH  Google Scholar 

  49. Shkoller, S.: Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations. Appl. Mat. Lett. 14(5), 539–543, 2001

    MathSciNet  MATH  Google Scholar 

  50. Shnirelman, A.: Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4(5), 586–620, 1994

    MathSciNet  MATH  Google Scholar 

  51. Shnirelman, A.I.: The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid. Math. USSR-Sb. 56(1), 79–105, 1987

    Google Scholar 

  52. Sickel, W., Ullrich, T.: Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross. J. Approx. Theory161(2), 748–786, 2009

    MathSciNet  MATH  Google Scholar 

  53. Triebel, H.: Theory of Function Spaces. Birkhäuser, Boston 1983

    MATH  Google Scholar 

  54. Triebel, H.: Theory of Function Spaces II. Birkhäuser, Boston 1992

    MATH  Google Scholar 

  55. Triebel, H.: The Structure of Functions, Monographs in Mathematics, vol. 97. Birkhäuser Verlag, Basel 2001

    Google Scholar 

  56. Vizman, C.: Geodesic equations on diffeomorphism groups. Symmetry Integr. Geom. Methods Appl. 4, 30, 2008

    MathSciNet  MATH  Google Scholar 

  57. Washabaugh, P.: The SQG equation as a geodesic equation. Arch. Ration. Mech. Anal. 222(3), 1269–1284, 2016

    MathSciNet  MATH  Google Scholar 

  58. Wolibner, W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogene, incompressible, pendant un temps infiniment long. Math. Z. 37, 698–726, 1933

    MathSciNet  MATH  Google Scholar 

  59. Wunsch, M.: On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric. J. Nonlinear Math. Phys. 17(1), 7–11, 2010

    ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, USA

    Martin Bauer

  2. Department of Mathematics, Albert Ludwig University of Freiburg, Freiburg, Germany

    Philipp Harms

  3. Department of Mathematics, Brooklyn College and Graduate Center, City University of New York, New York City, USA

    Stephen C. Preston

Authors
  1. Martin Bauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philipp Harms
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephen C. Preston
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Bauer.

Additional information

Communicated by V. Šverák

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

MB was partially supported by NSF-Grant 1912037 (collaborative research in connection with NSF-Grant 1912030). MB was also supported by a first year assistant professor award of the Florida State University. PH was supported by the Freiburg Institute of Advances Studies in the form of a Junior Fellowship. SCP was partially supported by Simons Foundation Collaboration Grant for Mathematicians No. 318969. SCP was also supported by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bauer, M., Harms, P. & Preston, S.C. Vanishing Distance Phenomena and the Geometric Approach to SQG. Arch Rational Mech Anal 235, 1445–1466 (2020). https://doi.org/10.1007/s00205-019-01449-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-019-01449-7

Search

Navigation