July  2013, 33(7): 2791-2808. doi: 10.3934/dcds.2013.33.2791

Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations

1. 

Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

2. 

UPMC Univ Paris 06, UMR 7598, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  March 2012 Revised  July 2012 Published  January 2013

It was recently proven by De Lellis, Kappeler, and Topalov in [19] that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space $Lip (\mathbb{T})$ endowed with the topology of $H^1 (\mathbb{T})$. We prove here that the Lagrangian flow of these solutions are analytic with respect to time and smooth with respect to the initial data.
    These results can be adapted to the higher-order Camassa-Holm equations describing the exponential curves of the manifold of orientation preserving diffeomorphisms of $\mathbb{T}$ using the Riemannian structure induced by the Sobolev inner product $H^l (\mathbb{T})$, for $l ∈ \mathbb{N}$, $l\geq 2$ (the classical Camassa-Holm equation corresponds to the case $l=1$): the periodic Cauchy problem is locally well-posed in the space $ W^{2l-1,\infty} (\mathbb{T})$ endowed with the topology of $H^{2l-1} (\mathbb{T})$ and the Lagrangian flows of these solutions are analytic with respect to time with values in $ W^{2l-1,\infty} (\mathbb{T})$ and smooth with respect to the initial data.
    These results extend some earlier results which dealt with more regular solutions. In particular our results cover the case of peakons, up to the first collision.
Citation: Olivier Glass, Franck Sueur. Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2791-2808. doi: 10.3934/dcds.2013.33.2791
References:
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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.   Google Scholar

[2]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces,, Int. Math. Res. Not., ().  doi: 10.1093/imrn/rnr218.  Google Scholar

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A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl. (Singap.), 5 (2007), 1.  doi: 10.1142/S0219530507000857.  Google Scholar

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A. Bressan and M. Fonte, An optimal transportation metric for solutions of the Camassa-Holm equation,, Methods Appl. Anal., 12 (2005), 191.   Google Scholar

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J.-Y. Chemin, Fluides parfaits incompressibles,, Astérisque, 230 (1995).   Google Scholar

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G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations,, Journal of Differential Equations, 246 (2009), 929.  doi: 10.1016/j.jde.2008.04.014.  Google Scholar

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A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

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A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303.   Google Scholar

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A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.  doi: 10.1007/BF02392586.  Google Scholar

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A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math. (2), 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

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A. Constantin and B. Kolev, $H^k$ metrics on the diffeomorphism group of the circle,, J. Nonlinear Math. Phys., 10 (2003), 424.  doi: 10.2991/jnmp.2003.10.4.1.  Google Scholar

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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.  Google Scholar

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A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C.  Google Scholar

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R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.   Google Scholar

[18]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429.  doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar

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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.  Google Scholar

[20]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math., 92 (1970), 102.   Google Scholar

[21]

K. El Dika and L. Molinet, Exponential decay of $H^1$-localized solutions and stability of the train of $N$ solitary waves for the Camassa-Holm equation,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2313.  doi: 10.1098/rsta.2007.2011.  Google Scholar

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K. El Dika and L. Molinet, Stability of multipeakons,, Ann. Inst. H. Poincaré Anal. Non Linaire, 26 (2009), 1517.  doi: 10.1016/j.anihpc.2009.02.002.  Google Scholar

[23]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (1981), 47.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[24]

O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation,, J. Differential Equations, 245 (2008), 1584.  doi: 10.1016/j.jde.2008.06.016.  Google Scholar

[25]

O. Glass, F. Sueur and T. Takahashi, Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid,, Annales Scientifiques de l'Ecole Normale Supérieure, 45 (2012), 1.   Google Scholar

[26]

H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view,, Comm. Partial Differential Equations, 32 (2007), 1511.  doi: 10.1080/03605300601088674.  Google Scholar

[27]

H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation,, Ann. Inst. Fourier (Grenoble), 58 (2008), 945.   Google Scholar

[28]

T. Kappeler, E. Loubet and P. Topalov, Analyticity of Riemannian exponential maps on Diff(T),, J. Lie Theory, 17 (2007), 481.   Google Scholar

[29]

T. Kato, On the smoothness of trajectories in incompressible perfect fluids,, in, 263 (2000), 109.  doi: 10.1090/conm/263/04194.  Google Scholar

[30]

J. Lenells, Stability of periodic peakons,, Int. Math. Res. Not., (2004), 485.  doi: 10.1155/S1073792804132431.  Google Scholar

[31]

J. Lenells, Riemannian geometry on the diffeomorphism group of the circle,, Ark. Mat., 45 (2007), 297.  doi: 10.1007/s11512-007-0047-8.  Google Scholar

[32]

Yi A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Differential Equations, 162 (2000), 27.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[33]

R. McLachlan and X. Zhang, Well-posedness of modified Camassa-Holm equations,, J. Differential Equations, 246 (2009), 3241.  doi: 10.1016/j.jde.2009.01.039.  Google Scholar

[34]

G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080.  doi: 10.1007/PL00012648.  Google Scholar

[35]

C. Mu, S. Zhou and R. Zeng, Well-posedness and blow-up phenomena for a higher order shallow water equation,, J. Differential Equations, 251 (2011), 3488.  doi: 10.1016/j.jde.2011.08.020.  Google Scholar

[36]

V. Perrollaz, Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval,, J. Funct. Anal., 259 (2010), 2333.  doi: 10.1016/j.jfa.2010.06.007.  Google Scholar

[37]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal. Ser. A: Theory Methods, 46 (2001), 309.  doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[38]

P. Serfati, Structures holomorphes à faible régularité spatiale en mécanique des fluides,, J. Math. Pures Appl., 74 (1995), 95.   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.   Google Scholar

[2]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces,, Int. Math. Res. Not., ().  doi: 10.1093/imrn/rnr218.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl. (Singap.), 5 (2007), 1.  doi: 10.1142/S0219530507000857.  Google Scholar

[4]

A. Bressan and M. Fonte, An optimal transportation metric for solutions of the Camassa-Holm equation,, Methods Appl. Anal., 12 (2005), 191.   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]

J.-Y. Chemin, Fluides parfaits incompressibles,, Astérisque, 230 (1995).   Google Scholar

[7]

G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations,, Journal of Differential Equations, 246 (2009), 929.  doi: 10.1016/j.jde.2008.04.014.  Google Scholar

[8]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303.   Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.  doi: 10.1007/BF02392586.  Google Scholar

[11]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math. (2), 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[12]

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

[13]

A. Constantin and B. Kolev, $H^k$ metrics on the diffeomorphism group of the circle,, J. Nonlinear Math. Phys., 10 (2003), 424.  doi: 10.2991/jnmp.2003.10.4.1.  Google Scholar

[14]

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.  Google Scholar

[15]

A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C.  Google Scholar

[16]

H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,, Acta Mech., 127 (1998), 193.  doi: 10.1007/BF01170373.  Google Scholar

[17]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.   Google Scholar

[18]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429.  doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar

[19]

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.  Google Scholar

[20]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math., 92 (1970), 102.   Google Scholar

[21]

K. El Dika and L. Molinet, Exponential decay of $H^1$-localized solutions and stability of the train of $N$ solitary waves for the Camassa-Holm equation,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2313.  doi: 10.1098/rsta.2007.2011.  Google Scholar

[22]

K. El Dika and L. Molinet, Stability of multipeakons,, Ann. Inst. H. Poincaré Anal. Non Linaire, 26 (2009), 1517.  doi: 10.1016/j.anihpc.2009.02.002.  Google Scholar

[23]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (1981), 47.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[24]

O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation,, J. Differential Equations, 245 (2008), 1584.  doi: 10.1016/j.jde.2008.06.016.  Google Scholar

[25]

O. Glass, F. Sueur and T. Takahashi, Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid,, Annales Scientifiques de l'Ecole Normale Supérieure, 45 (2012), 1.   Google Scholar

[26]

H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view,, Comm. Partial Differential Equations, 32 (2007), 1511.  doi: 10.1080/03605300601088674.  Google Scholar

[27]

H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation,, Ann. Inst. Fourier (Grenoble), 58 (2008), 945.   Google Scholar

[28]

T. Kappeler, E. Loubet and P. Topalov, Analyticity of Riemannian exponential maps on Diff(T),, J. Lie Theory, 17 (2007), 481.   Google Scholar

[29]

T. Kato, On the smoothness of trajectories in incompressible perfect fluids,, in, 263 (2000), 109.  doi: 10.1090/conm/263/04194.  Google Scholar

[30]

J. Lenells, Stability of periodic peakons,, Int. Math. Res. Not., (2004), 485.  doi: 10.1155/S1073792804132431.  Google Scholar

[31]

J. Lenells, Riemannian geometry on the diffeomorphism group of the circle,, Ark. Mat., 45 (2007), 297.  doi: 10.1007/s11512-007-0047-8.  Google Scholar

[32]

Yi A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Differential Equations, 162 (2000), 27.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[33]

R. McLachlan and X. Zhang, Well-posedness of modified Camassa-Holm equations,, J. Differential Equations, 246 (2009), 3241.  doi: 10.1016/j.jde.2009.01.039.  Google Scholar

[34]

G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080.  doi: 10.1007/PL00012648.  Google Scholar

[35]

C. Mu, S. Zhou and R. Zeng, Well-posedness and blow-up phenomena for a higher order shallow water equation,, J. Differential Equations, 251 (2011), 3488.  doi: 10.1016/j.jde.2011.08.020.  Google Scholar

[36]

V. Perrollaz, Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval,, J. Funct. Anal., 259 (2010), 2333.  doi: 10.1016/j.jfa.2010.06.007.  Google Scholar

[37]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal. Ser. A: Theory Methods, 46 (2001), 309.  doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[38]

P. Serfati, Structures holomorphes à faible régularité spatiale en mécanique des fluides,, J. Math. Pures Appl., 74 (1995), 95.   Google Scholar

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