The geometry of a vorticity model equation

Communications on Pure and Applied Analysis

Volumn 11, Issue 4, 2012, Pages 1407-1419

  Escher, Joachim ^a   Kolev, Boris ^b   Wunsch, Marcus ^c  

^a LEIBNIZ UNIVERSITY HANNOVER   (Germany)

^b CNRS   (France)

^c ETH ZURICH   (Switzerland)

Author keywords

CLM equation; Euler equation on diffeomorphisms group of the circle

Indexed keywords

EID: 84861936438     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2012.11.1407     Document Type: Article

References (36)

1. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71(1993), 1661-1664.
2. A. Castro and D. Córdoba, Infinite energy solutions of the surface quasi-geostrophic equation, Advances in Mathematics, 225(2010), 1820-1829.
3. A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35(2002), R51-R79. (Pubitemid 37049325)
4. A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78(2003), 787-804. (Pubitemid 37358018)
5. A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192(2009), 165-186.
6. P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38(1985), 715-724.
7. A. Cordoba, D. Cordoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162(2005), 1377-1389. (Pubitemid 44047888)
8. A. Cordoba, D. Cordoba and M. A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl., 86, 529-540.
9. A. Degasperis and M. Procesi, Asymptotic integrability, in "Symmetry and Perturbation Theory", Rome (1998) World Sci. Publishing, River Edge, NJ (1999) 23-37.
10. S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Statist. Phys., 59(1990), 1251-1263.
11. C. De Lellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation, Comm. Partial Differential Equations, 32(2007), 87-126.
12. N. Dunford and J. T. Schwartz, Linear operators. Part I, Wiley Classics Library. John Wiley & Sons Inc., New York, 1988. General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication.
13. D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math., 92(1970), 102-163.
14. J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Mathematische Zeitschrift, pages 1-17, 2010. 10.1007/s00209-010-0778-2.
15. J. Escher and B. Kolev, Right invariant Sobolev metrics Hs on the diffeomorphisms group of the circle, preprint 2011.
16. J. Escher and J. Seiler, The periodic b-equation and Euler equations on the circle, J. Math. Phys., 51 (2010).
17. J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, preprint 2010, arXiv:1009.1029.
18. L. Euler, Principes generaux du mouvement des fluides, Memoires de l'academie des sciences de Berlin, 11(1757), 274-315.
19. F. Gay-Balmaz, Infinite dimensional geodesic flows and the universal Teichmuller space, PhD thesis, Ecole Polytechnique Federale de Lausanne, Lausanne, 2009.
20. L. Guieu and C. Roger, L'algebre et le groupe de Virasoro, Les Publications CRM, Montreal, QC, 2007, Aspects geometriques et algebriques, generalisations. [Geometric and algebraic aspects, generalizations], With an appendix by Vlad Sergiescu.
21. R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N. S.), 7(1982), 165-222.
22. J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51(1991), 1498-1521.
23. B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176(2003), 116-144. (Pubitemid 36610186)
24. B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation, Wave Motion, 46(2009), 412-419.
25. J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57(2007), 2049-2064.
26. J. Lenells, The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal., 40(2008), 266-277.
27. H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21(2008), 2447-2461.
28. J. N. Pandey, "The Hilbert Transform of Schwartz Distributions and Applications", Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1996. A Wiley-Interscience Publication.
29. I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point, J. Fluid Mech., 12(1962), 161-168.
30. T. Sakajo, Blow-up solutions of the Constantin-Lax-Majda equation with a generalized viscosity term, J. Math. Sci. Univ. Tokyo, 10(2003), 187-207.
31. T. Sakajo, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term, Nonlinearity, 16(2003), 1319-1328.
32. S. Schochet, Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math., 39(1986), 531-537.
33. L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmuller space, Mem. Amer. Math. Soc, 183(2006), viii+119.
34. F. Tiǧlay and C. Vizman, Generalized Euler-Poincare equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys., 97(2011), 45-60.
35. E. Wegert and A. S. Vasudeva Murthy, Blow-up in a modified Constantin-Lax-Majda model for the vorticity equation, Z. Anal. Anwendungen, 18(1999), 183-191.
36. M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric, J. Nonlinear Math. Phys., 17(2010), 7-11.