The geometry of a vorticity model equation

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July 2012, 11(4): 1407-1419. doi: 10.3934/cpaa.2012.11.1407

The geometry of a vorticity model equation

Joachim Escher ^1, , Boris Kolev ^2, and Marcus Wunsch ^3,

1.	Institute for Applied Mathematics, University of Hanover, D-30167 Hanover
2.	LATP, CNRS & University of Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13
3.	Swiss Federal Institute of Technology Zurich, Department of Mathematics, 8092 Zurich

Received July 2011 Revised September 2011 Published January 2012

Abstract
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We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$ of orientation-preserving diffeomorphisms $\varphi \in \mathrm{Diff}^{\infty}(\mathbb{S})$ such that $\varphi(1) = 1$ equipped with the right-invariant metric induced by the homogeneous Sobolev norm $\dot H^{1/2}$. On the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$, this induces a weak Riemannian structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.

Keywords: Euler equation on diffeomorphisms group of the circle., CLM equation.

Mathematics Subject Classification: Primary: 58D05, 58B25; Secondary: 35Q35, 37K6.

Citation: Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407

References:

[1]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[2]	A. Castro and D. Córdoba, Infinite energy solutions of the surface quasi-geostrophic equation, Advances in Mathematics, 225 (2010), 1820-1829. doi: 10.1016/j.aim.2010.04.018.
[3]	A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201.
[4]	A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.
[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. doi: 10.1007/s00205-008-0128-2.
[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. doi: 10.1002/cpa.3160380605.
[7]	A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1377-1389. doi: 10.4007/annals.2005.162.1377.
[8]	A. Córdoba, D. Córdoba and M. A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl., 86, 529-540. doi: 10.1016/j.matpur.2006.08.002.
[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. doi: 10.1007/BF01334750.
[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. doi: 10.1080/03605300601091470.
[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. doi: 10.2307/1970699.
[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. doi: 10.1007/s00209-010-0778-2.
[15]	J. Escher and B. Kolev, Right invariant Sobolev metrics $H^s$ 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 généraux du mouvement des fluides, Mémoires de l'académie des sciences de Berlin, 11 (1757), 274-315.
[19]	F. Gay-Balmaz, Infinite dimensional geodesic flows and the universal Teichmüller space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009.
[20]	L. Guieu and C. Roger, L'algèbre et le groupe de Virasoro, Les Publications CRM, Montreal, QC, 2007, Aspects géométriques et algébriques, généralisations. [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. doi: 10.1137/0151075.
[23]	B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.
[24]	B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation, Wave Motion, 46 (2009), 412-419. doi: 10.1016/j.wavemoti.2009.06.005.
[25]	J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064. doi: 10.1016/j.geomphys.2007.05.003.
[26]	J. Lenells, The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277. doi: http://dx.doi.org/10.1137/050647451.
[27]	H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21 (2008), 2447-2461. doi: 10.1088/0951-7715/21/10/013.
[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. doi: 10.1017/S0022112062000130.
[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. doi: 10.1088/0951-7715/16/4/307.
[32]	S. Schochet, Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math., 39 (1986), 531-537. doi: 10.1002/cpa.3160390404.
[33]	L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc., 183 (2006), viii+119.
[34]	F. Tıǧlay and C. Vizman, Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys., 97 (2011), 45-60. doi: 10.1007/s11005-011-0464-2.
[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. doi: 10.1142/S1402925110000544.

show all references

References:

[1]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[2]	A. Castro and D. Córdoba, Infinite energy solutions of the surface quasi-geostrophic equation, Advances in Mathematics, 225 (2010), 1820-1829. doi: 10.1016/j.aim.2010.04.018.
[3]	A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201.
[4]	A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.
[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. doi: 10.1007/s00205-008-0128-2.
[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. doi: 10.1002/cpa.3160380605.
[7]	A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1377-1389. doi: 10.4007/annals.2005.162.1377.
[8]	A. Córdoba, D. Córdoba and M. A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl., 86, 529-540. doi: 10.1016/j.matpur.2006.08.002.
[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. doi: 10.1007/BF01334750.
[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. doi: 10.1080/03605300601091470.
[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. doi: 10.2307/1970699.
[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. doi: 10.1007/s00209-010-0778-2.
[15]	J. Escher and B. Kolev, Right invariant Sobolev metrics $H^s$ 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 généraux du mouvement des fluides, Mémoires de l'académie des sciences de Berlin, 11 (1757), 274-315.
[19]	F. Gay-Balmaz, Infinite dimensional geodesic flows and the universal Teichmüller space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009.
[20]	L. Guieu and C. Roger, L'algèbre et le groupe de Virasoro, Les Publications CRM, Montreal, QC, 2007, Aspects géométriques et algébriques, généralisations. [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. doi: 10.1137/0151075.
[23]	B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math., 176 (2003), 116-144. doi: 10.1016/S0001-8708(02)00063-4.
[24]	B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation, Wave Motion, 46 (2009), 412-419. doi: 10.1016/j.wavemoti.2009.06.005.
[25]	J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064. doi: 10.1016/j.geomphys.2007.05.003.
[26]	J. Lenells, The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277. doi: http://dx.doi.org/10.1137/050647451.
[27]	H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21 (2008), 2447-2461. doi: 10.1088/0951-7715/21/10/013.
[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. doi: 10.1017/S0022112062000130.
[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. doi: 10.1088/0951-7715/16/4/307.
[32]	S. Schochet, Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math., 39 (1986), 531-537. doi: 10.1002/cpa.3160390404.
[33]	L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc., 183 (2006), viii+119.
[34]	F. Tıǧlay and C. Vizman, Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications, Lett. Math. Phys., 97 (2011), 45-60. doi: 10.1007/s11005-011-0464-2.
[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. doi: 10.1142/S1402925110000544.

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