Abstract
We show that two families of equations, the generalized inviscid Proudman–Johnson equation and the r-Hunter–Saxton equation (recently introduced by Cotter et al.), coincide for a certain range of parameters. This gives a new geometric interpretation of these Proudman–Johnson equations as geodesic equations of right invariant homogeneous \(W^{1,r}\)-Finsler metrics on the diffeomorphism group. Generalizing a construction of Lenells for the Hunter–Saxton equation, we analyze these equations using an isometry from the diffeomorphism group to an appropriate subset of real-valued functions. Thereby, we show that the periodic case is equivalent to the geodesic equations on the \(L^r\)-sphere in the space of functions, and the non-periodic case is equivalent to a geodesic flow on a flat space. This allows us to give explicit solutions to these equations in the non-periodic case, and answer several questions of Cotter et al. regarding their limiting behavior.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
29 June 2023
A Correction to this paper has been published: https://doi.org/10.1007/s00332-023-09918-w
Notes
See also Escher and Kolev (2011) for a similar interpretation of the b-equations, which include the Camassa–Holm and the Degasperis–Procesi equation.
For \(\uplambda =0\), the integrability of the periodic \(\uplambda \)-PJ equation is shown in the article (Lenells and Misiołek 2014). This proof translates directly to the non-periodic case.
It was also used as a tool for studying the diameter of diffeomorphism groups (Bauer and Maor 2020).
References
Arnold, V.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Annales de l’institut Fourier 16(1), 319–361 (1966)
Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics, vol. 125. Springer, Berlin (1999)
Bauer, M., Maor, C.: Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics. to appear in Calc. Var. and PDEs (2020)
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 (2014a)
Bauer, M., Bruveris, M., Michor, P.W.: Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vis. 50(1–2), 60–97 (2014b)
Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661 (1993)
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)
Cotter, C.J., Deasy, J., Pryer, T.: The r-Hunter-Saxton equation, smooth and singular solutions and their approximation. Nonlinearity 33(12), 7016 (2020)
Escher, J., Kolev, B.: The Degasperis–Procesi equation as a non-metric Euler equation. Math. Z. 269(3–4), 1137–1153 (2011)
Escher, J., Kolev, B., Wunsch, M.: The geometry of a vorticity model equation. Commun. Pure Appl. Anal. 11(4), 1407 (2010)
Gibilisco, P.: \(L^p\) unit spheres and the \(\alpha \)-geometries: questions and perspectives. Entropy 22(12), 1409 (2020)
Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51(6), 1498–1521 (1991)
Kogelbauer, F.: On the global well-posedness of the inviscid generalized Proudman–Johnson equation using flow map arguments. J. Differ. Equ. 268(3), 1050–1080 (2020)
Kouranbaeva, S.: The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40(2), 857–868 (1999)
Kriegl, A., Michor, P.W., Rainer, A.: An exotic zoo of diffeomorphism groups on \({\mathbb{R}}^n\). Ann. Glob. Anal. Geom. 47(2), 179–222 (2015)
Lenells, J.: The Hunter-Saxton equation describes the geodesic flow on a sphere. J. Geom. Phys. 57(10), 2049–2064 (2007)
Lenells, J.: The Hunter–Saxton equation: a geometric approach. SIAM J. Math. Anal. 40(1), 266–277 (2008)
Lenells, J., Misiołek, G.: Amari–Chentsov connections and their geodesics on homogeneous spaces of diffeomorphism groups. J. Math. Sci. 196(2), 144–151 (2014)
Marquis, T., Neeb, K.-H.: Half-Lie groups. Transform. Groups 23(3), 801–840 (2018)
Michor, P.W., Mumford, D.: A zoo of diffeomorphism groups on \({\mathbb{R}}^n\). Ann. Glob. Anal. Geom. 4(44), 529–540 (2013)
Misiołek, G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24(3), 203–208 (1998)
Okamoto, H., Zhu, J.: Some similarity solutions of the Navier–Stokes equations and related topics. Taiwan. J. Math. 4(1), 65–103 (2000)
Omori, H.: On the group of diffeomorphisms on a compact manifold. In: Proceedings of the Symposium on the Pure and Applied Mathematics, XV, Amer. Math. Soc, pp. 167–183 (1970)
Ovsienko, V.Y., Khesin, B.A.: Korteweg–de Vries superequation as an Euler equation. Funct. Anal. Appl. 21(4), 329–331 (1987)
Sarria, A., Saxton, R.: Blow-up of solutions to the generalized inviscid Proudman–Johnson equation. J. Math. Fluid Mech. 15(3), 493–523 (2013)
Vizman, C., et al.: Geodesic equations on diffeomorphism groups. Symmetry Integr. Geom. Methods Appl. 4, 030 (2008)
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)
Wunsch, M.: The generalized Proudman–Johnson equation revisited. J. Math. Fluid Mech. 13(1), 147–154 (2011)
Acknowledgements
The authors are grateful to S. Preston and G. Misiołek for various discussions during the preparation of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
M. Bauer was partially supported by NSF-Grants 1912037 and 1953244. Y. Lu was partially supported by NSF-Grant 1912037. C. Maor was partially supported by ISF-Grant 1269/19.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bauer, M., Lu, Y. & Maor, C. A Geometric View on the Generalized Proudman–Johnson and r-Hunter–Saxton Equations. J Nonlinear Sci 32, 17 (2022). https://doi.org/10.1007/s00332-021-09775-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00332-021-09775-5