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

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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), 319-361. doi: 10.5802/aif.233.
[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), 5-21. doi: 10.1007/s10455-012-9353-x.
[3]	A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. doi: 10.1137/050623036.
[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), 385-395. doi: 10.1007/s10455-012-9350-0.
[5]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[6]	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.
[7]	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.
[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), 715-724. doi: 10.1002/cpa.3160380605.
[9]	D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.
[10]	J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153. doi: 10.1007/s00209-010-0778-2.
[11]	J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation, Commun. Pure Appl. Anal., 11 (2012), 1407-1419. doi: 10.3934/cpaa.2012.11.1407.
[12]	J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, Communications in Contemporary Mathematics, 14 (2012), 1250016, 13pp. URL http://adsabs.harvard.edu/abs/2010arXiv1009.1029E. doi: 10.1142/S0219199712500162.
[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), 154-193, URL http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1758&seite:int=156.
[14]	A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X.
[15]	F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009.
[16]	E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian geometry on infinite-dimensional manifolds, ArXiv e-prints, URL http://adsabs.harvard.edu/abs/2012arXiv1201.2251G. doi: 10.1007/s12220-014-9523-0.
[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, URL http://adsabs.harvard.edu/abs/2012arXiv1202.6135G.
[18]	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.
[19]	R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.
[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), 1-81. doi: 10.1006/aima.1998.1721.
[21]	J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.
[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, American Mathematical Society, 2013. doi: 10.1090/S0065-9266-2013-00676-4.
[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), 617-656. doi: 10.1007/s00208-008-0250-3.
[24]	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.
[25]	B. Kolev, Lie groups and mechanics: An introduction, J. Nonlinear Math. Phys., 11 (2004), 480-498. doi: 10.2991/jnmp.2004.11.4.5.
[26]	S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1999.
[27]	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.
[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), 129-161. doi: 10.1007/s00220-010-1069-9.
[29]	P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245 (electronic).
[30]	J. Milnor, Remarks on infinite-dimensional Lie groups, in Relativity, groups and topology, II (Les Houches, 1983), North-Holland, Amsterdam, (1984), 1007-1057.
[31]	G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.
[32]	G. Misiołek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104. doi: 10.1007/PL00012648.
[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), 1-34, URL http://projecteuclid.org/getRecord?id=euclid.ojm/1200785862.
[34]	P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.
[35]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.
[36]	H. Poincaré, Sur une nouvelle forme des équations de la mécanique, C.R. Acad. Sci., 132 (1901), 369-371.
[37]	S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160 (1998), 337-365. doi: 10.1006/jfan.1998.3335.
[38]	L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc., 183 (2006), viii+119pp. doi: 10.1090/memo/0861.
[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), 45-60. doi: 10.1007/s11005-011-0464-2.
[40]	H. Triebel, Theory of Function Spaces, Birkhäuser Boston, 1983. doi: 10.1007/978-3-0346-0416-1.
[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), 7-11. doi: 10.1142/S1402925110000544.
[42]	Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283 (electronic). doi: 10.1137/S0036141003425672.

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), 319-361. doi: 10.5802/aif.233.
[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), 5-21. doi: 10.1007/s10455-012-9353-x.
[3]	A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. doi: 10.1137/050623036.
[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), 385-395. doi: 10.1007/s10455-012-9350-0.
[5]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[6]	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.
[7]	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.
[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), 715-724. doi: 10.1002/cpa.3160380605.
[9]	D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.
[10]	J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153. doi: 10.1007/s00209-010-0778-2.
[11]	J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation, Commun. Pure Appl. Anal., 11 (2012), 1407-1419. doi: 10.3934/cpaa.2012.11.1407.
[12]	J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation, Communications in Contemporary Mathematics, 14 (2012), 1250016, 13pp. URL http://adsabs.harvard.edu/abs/2010arXiv1009.1029E. doi: 10.1142/S0219199712500162.
[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), 154-193, URL http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1758&seite:int=156.
[14]	A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X.
[15]	F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 2009.
[16]	E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian geometry on infinite-dimensional manifolds, ArXiv e-prints, URL http://adsabs.harvard.edu/abs/2012arXiv1201.2251G. doi: 10.1007/s12220-014-9523-0.
[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, URL http://adsabs.harvard.edu/abs/2012arXiv1202.6135G.
[18]	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.
[19]	R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.
[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), 1-81. doi: 10.1006/aima.1998.1721.
[21]	J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.
[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, American Mathematical Society, 2013. doi: 10.1090/S0065-9266-2013-00676-4.
[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), 617-656. doi: 10.1007/s00208-008-0250-3.
[24]	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.
[25]	B. Kolev, Lie groups and mechanics: An introduction, J. Nonlinear Math. Phys., 11 (2004), 480-498. doi: 10.2991/jnmp.2004.11.4.5.
[26]	S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1999.
[27]	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.
[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), 129-161. doi: 10.1007/s00220-010-1069-9.
[29]	P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10 (2005), 217-245 (electronic).
[30]	J. Milnor, Remarks on infinite-dimensional Lie groups, in Relativity, groups and topology, II (Les Houches, 1983), North-Holland, Amsterdam, (1984), 1007-1057.
[31]	G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.
[32]	G. Misiołek, Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12 (2002), 1080-1104. doi: 10.1007/PL00012648.
[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), 1-34, URL http://projecteuclid.org/getRecord?id=euclid.ojm/1200785862.
[34]	P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.
[35]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.
[36]	H. Poincaré, Sur une nouvelle forme des équations de la mécanique, C.R. Acad. Sci., 132 (1901), 369-371.
[37]	S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160 (1998), 337-365. doi: 10.1006/jfan.1998.3335.
[38]	L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space, Mem. Amer. Math. Soc., 183 (2006), viii+119pp. doi: 10.1090/memo/0861.
[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), 45-60. doi: 10.1007/s11005-011-0464-2.
[40]	H. Triebel, Theory of Function Spaces, Birkhäuser Boston, 1983. doi: 10.1007/978-3-0346-0416-1.
[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), 7-11. doi: 10.1142/S1402925110000544.
[42]	Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283 (electronic). doi: 10.1137/S0036141003425672.

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