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March  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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[12]

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

[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}. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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}. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[42]

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

[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}. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[12]

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

[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}. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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}. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

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

[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. Google Scholar

[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. Google Scholar

[42]

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

[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}. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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