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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1016/j.matpur.2006.08.002.  Google Scholar

[9]

A. Degasperis and M. Procesi, Asymptotic integrability, in "Symmetry and Perturbation Theory,'' Rome (1998) World Sci. Publishing, River Edge, NJ (1999) 23-37.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Escher and B. Kolev, Right invariant Sobolev metrics $H^s$ on the diffeomorphisms group of the circle, preprint 2011. Google Scholar

[16]

J. Escher and J. Seiler, The periodic $b$-equation and Euler equations on the circle, J. Math. Phys., 51 (2010).  Google Scholar

[17]

J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, preprint 2010, arXiv:1009.1029. Google Scholar

[18]

L. Euler, Principes généraux du mouvement des fluides, Mémoires de l'académie des sciences de Berlin, 11 (1757), 274-315. Google Scholar

[19]

F. Gay-Balmaz, Infinite dimensional geodesic flows and the universal Teichmüller space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009. Google Scholar

[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.  Google Scholar

[21]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 165-222.  Google Scholar

[22]

J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

S. Schochet, Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math., 39 (1986), 531-537. doi: 10.1002/cpa.3160390404.  Google Scholar

[33]

L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc., 183 (2006), viii+119.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1016/j.matpur.2006.08.002.  Google Scholar

[9]

A. Degasperis and M. Procesi, Asymptotic integrability, in "Symmetry and Perturbation Theory,'' Rome (1998) World Sci. Publishing, River Edge, NJ (1999) 23-37.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Escher and B. Kolev, Right invariant Sobolev metrics $H^s$ on the diffeomorphisms group of the circle, preprint 2011. Google Scholar

[16]

J. Escher and J. Seiler, The periodic $b$-equation and Euler equations on the circle, J. Math. Phys., 51 (2010).  Google Scholar

[17]

J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, preprint 2010, arXiv:1009.1029. Google Scholar

[18]

L. Euler, Principes généraux du mouvement des fluides, Mémoires de l'académie des sciences de Berlin, 11 (1757), 274-315. Google Scholar

[19]

F. Gay-Balmaz, Infinite dimensional geodesic flows and the universal Teichmüller space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009. Google Scholar

[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.  Google Scholar

[21]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 165-222.  Google Scholar

[22]

J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

S. Schochet, Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math., 39 (1986), 531-537. doi: 10.1002/cpa.3160390404.  Google Scholar

[33]

L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc., 183 (2006), viii+119.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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