Abstract
We consider the periodic μDP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection ∇ on the Fréchet Lie group Diff∞(\(\mathbb{S}^1\)) of all smooth and orientation-preserving diffeomorphisms of the circle \(\mathbb{S}^1\,=\,\mathbb{R}/\mathbb{Z}\). On the Lie algebra C∞(\(\mathbb{S}^1\)) of Diff∞(\(\mathbb{S}^1\)), this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of μDP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by ∇ is a smooth local diffeomorphism of a neighbourhood of zero in C∞(\(\mathbb{S}^1\)) onto a neighbourhood of the unit element in Diff∞(\(\mathbb{S}^1\)). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Fréchet space C∞(\(\mathbb{S}^1\)), and a sharp spatial regularity result for the geodesic flow.
Keywords
Mathematics Subject Classification (2000). Primary 53D25; Secondary 37K65.
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References
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.
V.I. Arnold, Mathematical Methods of Classical Mechanics. Springer, 1989.
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II: The KdV equation. Geom. Funct. Anal., 3 (1993), 209–262.
R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71 (1993), 1661–1664.
A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math., 51 (1998), 475–504.
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations. Acta Math., 181 (1998), 229–243.
A. Constantin and J. Escher, Global weak solutions for a shallow water equation. Indiana Univ. Math. J., 47 (1998), 1527–1545.
A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems. J. Phys. A, 35 (2002), R51–R79.
A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv., 78 (2003), 787–804.
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal., 192 (2009), 165–186.
C. De Lellis, T. Kappeler, and P. Topalov, Low regularity solutions of the periodic Camassa-Holm equation. Comm. in PDE, 32 (2007), 82–126.
A. Degasperis and M. Procesi, Asymptotic integrability. Symmetry and perturbation theory. (eds. A. Degasperis and G. Gaeta), Singapore:World Scientific (1999), 23–27.
A. Degasperis, D.D. Holm, and A.N.I. Hone, A new integrable equation with peakon solutions. Teoret. Mat. Fiz., 133 (2002), 170–183.
H.R. Dullin, G. Gottwald, and D.D. Holm, On asymptotically equivalent shallow water wave equations. Phys. D, 190 (2004), 1–14.
D.G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. of Math., 92 (1970), 102–163.
J. Escher, Wave breaking and shock waves for a periodic shallow water equation. Phil. Trans. R. Soc. A, 365 (2007), 2281–2289.
J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation, to appear in Math. Z., 2011.
J. Escher and J. Seiler, The periodic b-equation and Euler equations on the circle. J. Math. Phys., 51 (2010).
J. Escher and Z. Yin, Well-posedness, blow-up phenomena and global solutions for the b-equation. J. Reine Angew. Math., 624 (2008), 51–80.
J. Escher, Y. Liu, and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation. Indiana Univ. Math. J., 56 (2007), 87–117.
R.I. Ivanov.Water waves and integrability. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267–2280.
R.S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech., 455 (2007), 63–82.
T. Kappeler and P. Topalov, Global well-posedness of KdV in H 1(T;ℝ). Duke Math. J., 135 (2006), 327–360.
C.E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math., 46 (1993), 527–620.
B. Kolev, Some geometric investigations of the Degasperis-Procesi shallow water equation. Wave motion, 46 (2009), 412–419.
S. Kouranbaeva, The Camassa-Holm equation as a geodesic fow on the diffeomorphism group. J. Math. Phys., 40 (1999), 857–868.
S. Lang, Real and Functional Analysis, 3rd Edition, Springer, 1993.
S. Lang, Fundamentals of Differential Geometry, Springer, 1999.
J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere. J. Geom. Phys., 57 (2007), 2049–2064.
J. Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation. Disc. Cont. Dyn. Syst., 18 (2007), 643–656.
J. Lenells, G. Misiołek, and F. Tiğlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Commun. Math. Phys., 299 (2010), 129–161.
P.W. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach. Volume 69 of Progr. Nonlinear Differential Equations Appl., 133–215, Birkhäuser, 2006.
G. Misiołek, Classical solutions of the periodic Camassa-Holm equation. Geom. Funct. Anal., 12 (2002), 1080–1104.
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Dedicated to Herbert Amann on the occasion of his 70th birthday.
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Escher, J., Kohlmann, M., Kolev, B. (2011). Geometric Aspects of the Periodic μ-Degasperis-Procesi Equation. In: Escher, J., et al. Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 80. Springer, Basel. https://doi.org/10.1007/978-3-0348-0075-4_10
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DOI: https://doi.org/10.1007/978-3-0348-0075-4_10
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