Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

Joachim Escher; Boris Kolev

doi:10.3934/jgm.2014.6.335

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September 2014, 6(3): 335-372. doi: 10.3934/jgm.2014.6.335

Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

Joachim Escher ^1, and Boris Kolev ^2,

1.	Institute for Applied Mathematics, University of Hanover, D-30167 Hanover
2.	Institut de Mathématiques de Marseille, Aix Marseille Université, CNRS, Centrale Marseille, 13453 Marseille, France

Received November 2013 Revised March 2014 Published September 2014

Abstract
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In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.

Keywords: Sobolev metrics of fractional order., Euler equation, diffeomorphism group of the circle.

Mathematics Subject Classification: Primary: 58D05; Secondary: 35Q5.

Citation: Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335

References:

[1]	V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319. doi: 10.5802/aif.233. Google Scholar
[2]	M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Global Anal. Geom., 44 (2013), 5. doi: 10.1007/s10455-012-9353-x. Google Scholar
[3]	A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996. doi: 10.1137/050623036. Google Scholar
[4]	M. Bruveris, The energy functional on the Virasoro-Bott group with the $L^{2}$-metric has no local minima,, Ann. Global Anal. Geom., 43 (2013), 385. doi: 10.1007/s10455-012-9350-0. Google Scholar
[5]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[6]	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. Google Scholar
[7]	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. Google Scholar
[8]	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. Google Scholar
[9]	D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102. doi: 10.2307/1970699. Google Scholar
[10]	J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269 (2011), 1137. doi: 10.1007/s00209-010-0778-2. Google Scholar
[11]	J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation,, Commun. Pure Appl. Anal., 11 (2012), 1407. doi: 10.3934/cpaa.2012.11.1407. Google Scholar
[12]	J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Communications in Contemporary Mathematics, 14 (2012). doi: 10.1142/S0219199712500162. Google Scholar
[13]	L. P. Euler, Du mouvement de rotation des corps solides autour d'un axe variable,, Mémoires de l'académie des sciences de Berlin, 14 (1765), 154. Google Scholar
[14]	A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar
[15]	F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space,, PhD thesis, (2009). Google Scholar
[16]	E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian geometry on infinite-dimensional manifolds,, ArXiv e-prints, (). doi: 10.1007/s12220-014-9523-0. Google Scholar
[17]	E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve,, ArXiv e-prints, (). Google Scholar
[18]	L. Guieu and C. Roger, L'algèbre et le Groupe de Virasoro,, Les Publications CRM, (2007). Google Scholar
[19]	R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65. doi: 10.1090/S0273-0979-1982-15004-2. Google Scholar
[20]	D. D. Holm, J. E. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721. Google Scholar
[21]	J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075. Google Scholar
[22]	H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, vol. 226 of Memoirs of the American Mathematical Society,, 1st edition, (2013). doi: 10.1090/S0065-9266-2013-00676-4. Google Scholar
[23]	B. Khesin, J. Lenells and G. Misiołek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617. doi: 10.1007/s00208-008-0250-3. Google Scholar
[24]	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. Google Scholar
[25]	B. Kolev, Lie groups and mechanics: An introduction,, J. Nonlinear Math. Phys., 11 (2004), 480. doi: 10.2991/jnmp.2004.11.4.5. Google Scholar
[26]	S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics,, Springer-Verlag, (1999). Google Scholar
[27]	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. Google Scholar
[28]	J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129. doi: 10.1007/s00220-010-1069-9. Google Scholar
[29]	P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217. Google Scholar
[30]	J. Milnor, Remarks on infinite-dimensional Lie groups,, in Relativity, (1984), 1007. Google Scholar
[31]	G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar
[32]	G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080. doi: 10.1007/PL00012648. Google Scholar
[33]	S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle,, Osaka J. Math., 32 (1995), 1. Google Scholar
[34]	P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics,, 2nd edition, (1993). doi: 10.1007/978-1-4612-4350-2. Google Scholar
[35]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[36]	H. Poincaré, Sur une nouvelle forme des équations de la mécanique,, C.R. Acad. Sci., 132 (1901), 369. Google Scholar
[37]	S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics,, J. Funct. Anal., 160 (1998), 337. doi: 10.1006/jfan.1998.3335. Google Scholar
[38]	L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space,, Mem. Amer. Math. Soc., 183 (2006). doi: 10.1090/memo/0861. Google Scholar
[39]	F. Tiǧlay 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. Google Scholar
[40]	H. Triebel, Theory of Function Spaces,, Birkhäuser Boston, (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar
[41]	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. Google Scholar
[42]	Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672. Google Scholar

show all references

References:

[1]	V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319. doi: 10.5802/aif.233. Google Scholar
[2]	M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Global Anal. Geom., 44 (2013), 5. doi: 10.1007/s10455-012-9353-x. Google Scholar
[3]	A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996. doi: 10.1137/050623036. Google Scholar
[4]	M. Bruveris, The energy functional on the Virasoro-Bott group with the $L^{2}$-metric has no local minima,, Ann. Global Anal. Geom., 43 (2013), 385. doi: 10.1007/s10455-012-9350-0. Google Scholar
[5]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[6]	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. Google Scholar
[7]	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. Google Scholar
[8]	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. Google Scholar
[9]	D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102. doi: 10.2307/1970699. Google Scholar
[10]	J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269 (2011), 1137. doi: 10.1007/s00209-010-0778-2. Google Scholar
[11]	J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation,, Commun. Pure Appl. Anal., 11 (2012), 1407. doi: 10.3934/cpaa.2012.11.1407. Google Scholar
[12]	J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Communications in Contemporary Mathematics, 14 (2012). doi: 10.1142/S0219199712500162. Google Scholar
[13]	L. P. Euler, Du mouvement de rotation des corps solides autour d'un axe variable,, Mémoires de l'académie des sciences de Berlin, 14 (1765), 154. Google Scholar
[14]	A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar
[15]	F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space,, PhD thesis, (2009). Google Scholar
[16]	E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian geometry on infinite-dimensional manifolds,, ArXiv e-prints, (). doi: 10.1007/s12220-014-9523-0. Google Scholar
[17]	E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve,, ArXiv e-prints, (). Google Scholar
[18]	L. Guieu and C. Roger, L'algèbre et le Groupe de Virasoro,, Les Publications CRM, (2007). Google Scholar
[19]	R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65. doi: 10.1090/S0273-0979-1982-15004-2. Google Scholar
[20]	D. D. Holm, J. E. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1. doi: 10.1006/aima.1998.1721. Google Scholar
[21]	J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075. Google Scholar
[22]	H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, vol. 226 of Memoirs of the American Mathematical Society,, 1st edition, (2013). doi: 10.1090/S0065-9266-2013-00676-4. Google Scholar
[23]	B. Khesin, J. Lenells and G. Misiołek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617. doi: 10.1007/s00208-008-0250-3. Google Scholar
[24]	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. Google Scholar
[25]	B. Kolev, Lie groups and mechanics: An introduction,, J. Nonlinear Math. Phys., 11 (2004), 480. doi: 10.2991/jnmp.2004.11.4.5. Google Scholar
[26]	S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics,, Springer-Verlag, (1999). Google Scholar
[27]	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. Google Scholar
[28]	J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129. doi: 10.1007/s00220-010-1069-9. Google Scholar
[29]	P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217. Google Scholar
[30]	J. Milnor, Remarks on infinite-dimensional Lie groups,, in Relativity, (1984), 1007. Google Scholar
[31]	G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar
[32]	G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080. doi: 10.1007/PL00012648. Google Scholar
[33]	S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle,, Osaka J. Math., 32 (1995), 1. Google Scholar
[34]	P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics,, 2nd edition, (1993). doi: 10.1007/978-1-4612-4350-2. Google Scholar
[35]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[36]	H. Poincaré, Sur une nouvelle forme des équations de la mécanique,, C.R. Acad. Sci., 132 (1901), 369. Google Scholar
[37]	S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics,, J. Funct. Anal., 160 (1998), 337. doi: 10.1006/jfan.1998.3335. Google Scholar
[38]	L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space,, Mem. Amer. Math. Soc., 183 (2006). doi: 10.1090/memo/0861. Google Scholar
[39]	F. Tiǧlay 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. Google Scholar
[40]	H. Triebel, Theory of Function Spaces,, Birkhäuser Boston, (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar
[41]	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. Google Scholar
[42]	Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672. Google Scholar

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Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

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