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

The geometry of a vorticity model equation

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.
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. 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. 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). 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. 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. 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. 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. 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. doi: 10.1016/j.matpur.2006.08.002.

[9]

A. Degasperis and M. Procesi, Asymptotic integrability,, in, (1998), 23.

[10]

S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation,, J. Statist. Phys., 59 (1990), 1251. 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. doi: 10.1080/03605300601091470.

[12]

N. Dunford and J. T. Schwartz, Linear operators. Part I,, Wiley Classics Library. John Wiley & Sons Inc., (1988).

[13]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid,, Ann. of Math., 92 (1970), 102. doi: 10.2307/1970699.

[14]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Mathematische Zeitschrift, (2010), 1. 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., (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, (2010).

[18]

L. Euler, Principes généraux du mouvement des fluides,, M\'emoires de l'acad\'emie des sciences de Berlin, 11 (1757), 274.

[19]

F. Gay-Balmaz, Infinite dimensional geodesic flows and the universal Teichmüller space,, PhD thesis, (2009).

[20]

L. Guieu and C. Roger, L'algèbre et le groupe de Virasoro,, Les Publications CRM, (2007).

[21]

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

[22]

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

[23]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116. 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. 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. 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. 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. 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., (1996).

[29]

I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point,, J. Fluid Mech., 12 (1962), 161. 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.

[31]

T. Sakajo, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term,, Nonlinearity, 16 (2003), 1319. 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. 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).

[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. 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.

[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. 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. 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. 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). 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. 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. 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. 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. 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. doi: 10.1016/j.matpur.2006.08.002.

[9]

A. Degasperis and M. Procesi, Asymptotic integrability,, in, (1998), 23.

[10]

S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation,, J. Statist. Phys., 59 (1990), 1251. 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. doi: 10.1080/03605300601091470.

[12]

N. Dunford and J. T. Schwartz, Linear operators. Part I,, Wiley Classics Library. John Wiley & Sons Inc., (1988).

[13]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid,, Ann. of Math., 92 (1970), 102. doi: 10.2307/1970699.

[14]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Mathematische Zeitschrift, (2010), 1. 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., (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, (2010).

[18]

L. Euler, Principes généraux du mouvement des fluides,, M\'emoires de l'acad\'emie des sciences de Berlin, 11 (1757), 274.

[19]

F. Gay-Balmaz, Infinite dimensional geodesic flows and the universal Teichmüller space,, PhD thesis, (2009).

[20]

L. Guieu and C. Roger, L'algèbre et le groupe de Virasoro,, Les Publications CRM, (2007).

[21]

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

[22]

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

[23]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116. 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. 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. 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. 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. 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., (1996).

[29]

I. Proudman and K. Johnson, Boundary-layer growth near a rear stagnation point,, J. Fluid Mech., 12 (1962), 161. 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.

[31]

T. Sakajo, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term,, Nonlinearity, 16 (2003), 1319. 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. 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).

[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. 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.

[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. doi: 10.1142/S1402925110000544.

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