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2015, 35(3): 1327-1358. doi: 10.3934/dcds.2015.35.1327

On the initial value problem for higher dimensional Camassa-Holm equations

1. 

College of Information Science and Technology, Jinan University, Guangzhou, 510632, China

2. 

Department of Mathematics, Zhongshan University, Guangzhou, 510275

Received  April 2014 Revised  July 2014 Published  October 2014

This paper is concerned with the the initial value problem for higher dimensional Camassa-Holm equations. Firstly, the local well-posedness for this equations in both supercritical and critical Besov spaces are established. Then two blow-up criterions of strong solutions to the equations are derived. Finally, the analyticity of its solutions is proved in both variables, globally in space and locally in time.
Citation: Kai Yan, Zhaoyang Yin. On the initial value problem for higher dimensional Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1327-1358. doi: 10.3934/dcds.2015.35.1327
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der MathematischenWissenschaften, (2011). doi: 10.1007/978-3-642-16830-7.

[2]

S. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems,, Comm. Partial Differential Equations, 2 (1977), 1151. doi: 10.1080/03605307708820057.

[3]

R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy,, Adv. Math., 140 (1998), 190. doi: 10.1006/aima.1998.1768.

[4]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires,, Ann. Sci. École Norm. Sup., 14 (1981), 209.

[5]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z.

[6]

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

[7]

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.

[8]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0.

[9]

A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Four. (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757.

[10]

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

[11]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Annali Sc. Norm. Sup. Pisa., 26 (1998), 303.

[12]

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

[13]

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. doi: {10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5}.

[14]

A. Constantin and J. Escher, Particles trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[15]

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.

[16]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comm. Math. Helv., 78 (2003), 787. doi: 10.1007/s00014-003-0785-6.

[17]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2.

[18]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: 10.1007/s002200050801.

[19]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: {10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L}.

[20]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonlinear Sci., 12 (2002), 415. doi: 10.1007/s00332-002-0517-x.

[21]

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

[22]

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

[23]

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.

[24]

H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Letters, 87 (2001), 4501. doi: 10.1103/PhysRevLett.87.194501.

[25]

J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations,, J. Funct. Anal., 256 (2009), 479. doi: 10.1016/j.jfa.2008.07.010.

[26]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).

[27]

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

[28]

C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation,, Contemp. Math., 526 (2010), 199. doi: 10.1090/conm/526/10382.

[29]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 28 (2011), 623. doi: 10.1016/j.anihpc.2011.04.003.

[30]

A. A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow-water equation,, Differential Integral Equations, 14 (2001), 821.

[31]

A. A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575. doi: 10.1007/s00208-003-0466-1.

[32]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2 (2003), 323. doi: 10.1137/S1111111102410943.

[33]

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation,, The breadth of symplectic and Poisson geometry, 232 (2005), 203. doi: 10.1007/0-8176-4419-9_8.

[34]

D. D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two-component Camassa-Holm equation,, Phys. Rev. E(3), 79 (2009). doi: 10.1103/PhysRevE.79.016601.

[35]

R. S. Johnson and Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid. Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224.

[36]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Spectral Theory and Differential Equations, 448 (1975), 25.

[37]

M. Kohlmann, The two-dimensional periodic b-equation on the diffeomorphism group of the torus,, J. Phys. A, 44 (2011). doi: 10.1088/1751-8113/44/46/465205.

[38]

G. Misiolek, 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.

[39]

T. Nishida, A note on a theorem of Nirenberg,, J. Differential Geom., 12 (1977), 629.

[40]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X.

[41]

W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation,, J. Funct. Anal., 261 (2011), 1204. doi: 10.1016/j.jfa.2011.04.015.

[42]

G. B. Whitham, Linear and Nonlinear Waves,, J. Wiley & Sons, (1974).

[43]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: {10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5}.

[44]

K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems,, Math. Z., 269 (2011), 1113. doi: 10.1007/s00209-010-0775-5.

[45]

K. Yan and Z. Yin, Well-posedness for a modified two-component Camassa-Holm system in critical spaces,, Discrete Contin. Dyn. Syst., 33 (2013), 1699. doi: 10.3934/dcds.2013.33.1699.

[46]

K. Yan and Z. Yin, Initial boundary value problems for the two-component shallow water systems,, Rev. Mat. Iberoamericana, 29 (2013), 911. doi: 10.4171/RMI/744.

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der MathematischenWissenschaften, (2011). doi: 10.1007/978-3-642-16830-7.

[2]

S. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems,, Comm. Partial Differential Equations, 2 (1977), 1151. doi: 10.1080/03605307708820057.

[3]

R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy,, Adv. Math., 140 (1998), 190. doi: 10.1006/aima.1998.1768.

[4]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires,, Ann. Sci. École Norm. Sup., 14 (1981), 209.

[5]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z.

[6]

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

[7]

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.

[8]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0.

[9]

A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Four. (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757.

[10]

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

[11]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Annali Sc. Norm. Sup. Pisa., 26 (1998), 303.

[12]

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

[13]

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. doi: {10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5}.

[14]

A. Constantin and J. Escher, Particles trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[15]

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.

[16]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comm. Math. Helv., 78 (2003), 787. doi: 10.1007/s00014-003-0785-6.

[17]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2.

[18]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: 10.1007/s002200050801.

[19]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: {10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L}.

[20]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonlinear Sci., 12 (2002), 415. doi: 10.1007/s00332-002-0517-x.

[21]

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

[22]

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

[23]

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.

[24]

H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Letters, 87 (2001), 4501. doi: 10.1103/PhysRevLett.87.194501.

[25]

J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations,, J. Funct. Anal., 256 (2009), 479. doi: 10.1016/j.jfa.2008.07.010.

[26]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).

[27]

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

[28]

C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation,, Contemp. Math., 526 (2010), 199. doi: 10.1090/conm/526/10382.

[29]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 28 (2011), 623. doi: 10.1016/j.anihpc.2011.04.003.

[30]

A. A. Himonas and G. Misiolek, The Cauchy problem for an integrable shallow-water equation,, Differential Integral Equations, 14 (2001), 821.

[31]

A. A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575. doi: 10.1007/s00208-003-0466-1.

[32]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2 (2003), 323. doi: 10.1137/S1111111102410943.

[33]

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation,, The breadth of symplectic and Poisson geometry, 232 (2005), 203. doi: 10.1007/0-8176-4419-9_8.

[34]

D. D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two-component Camassa-Holm equation,, Phys. Rev. E(3), 79 (2009). doi: 10.1103/PhysRevE.79.016601.

[35]

R. S. Johnson and Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid. Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224.

[36]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Spectral Theory and Differential Equations, 448 (1975), 25.

[37]

M. Kohlmann, The two-dimensional periodic b-equation on the diffeomorphism group of the torus,, J. Phys. A, 44 (2011). doi: 10.1088/1751-8113/44/46/465205.

[38]

G. Misiolek, 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.

[39]

T. Nishida, A note on a theorem of Nirenberg,, J. Differential Geom., 12 (1977), 629.

[40]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X.

[41]

W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation,, J. Funct. Anal., 261 (2011), 1204. doi: 10.1016/j.jfa.2011.04.015.

[42]

G. B. Whitham, Linear and Nonlinear Waves,, J. Wiley & Sons, (1974).

[43]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: {10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5}.

[44]

K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems,, Math. Z., 269 (2011), 1113. doi: 10.1007/s00209-010-0775-5.

[45]

K. Yan and Z. Yin, Well-posedness for a modified two-component Camassa-Holm system in critical spaces,, Discrete Contin. Dyn. Syst., 33 (2013), 1699. doi: 10.3934/dcds.2013.33.1699.

[46]

K. Yan and Z. Yin, Initial boundary value problems for the two-component shallow water systems,, Rev. Mat. Iberoamericana, 29 (2013), 911. doi: 10.4171/RMI/744.

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