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Volumn 248, Issue 8, 2010, Pages 2038-2063

The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation

Author keywords

Generalized Camassa Holm equation; High order nonlinear terms; Local well posedness; Pseudoparabolic regularization method; Weak solution

Indexed keywords


EID: 76549131747     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2010.01.008     Document Type: Article
Times cited : (71)

References (48)
  • 1
    • 0001119719 scopus 로고
    • The initial value problem for the Korteweg-de Vries equation
    • Bona J., and Smith R. The initial value problem for the Korteweg-de Vries equation. Philos. Trans. R. Soc. London Ser. A 278 (1975) 555-601
    • (1975) Philos. Trans. R. Soc. London Ser. A , vol.278 , pp. 555-601
    • Bona, J.1    Smith, R.2
  • 2
    • 33846878957 scopus 로고    scopus 로고
    • Global conservative solutions of the Camassa-Holm equation
    • Bressan A., and Constantin A. Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. 183 (2007) 215-239
    • (2007) Arch. Ration. Mech. Anal. , vol.183 , pp. 215-239
    • Bressan, A.1    Constantin, A.2
  • 3
    • 51149109959 scopus 로고    scopus 로고
    • Global dissipative solutions of the Camassa-Holm equation
    • Bressan A., and Constantin A. Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. (Singap.) 5 (2007) 1-27
    • (2007) Anal. Appl. (Singap.) , vol.5 , pp. 1-27
    • Bressan, A.1    Constantin, A.2
  • 4
    • 12044254491 scopus 로고
    • An integrable shallow water equation with peaked solitons
    • Camassa R., and Holm D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993) 1661-1664
    • (1993) Phys. Rev. Lett. , vol.71 , pp. 1661-1664
    • Camassa, R.1    Holm, D.2
  • 7
    • 33746133952 scopus 로고    scopus 로고
    • Global weak solutions to a generalized hyperelastic-rod wave equation
    • Coclite G.M., Holden H., and Karlsen K.H. Global weak solutions to a generalized hyperelastic-rod wave equation. SIAM J. Math. Anal. 37 4 (2005) 1044-1069
    • (2005) SIAM J. Math. Anal. , vol.37 , Issue.4 , pp. 1044-1069
    • Coclite, G.M.1    Holden, H.2    Karlsen, K.H.3
  • 8
    • 33750526877 scopus 로고    scopus 로고
    • The trajectories of particles in Stokes waves
    • Constantin A. The trajectories of particles in Stokes waves. Invent. Math. 166 (2006) 523-535
    • (2006) Invent. Math. , vol.166 , pp. 523-535
    • Constantin, A.1
  • 9
    • 0034415062 scopus 로고    scopus 로고
    • Existence of permanent and breaking waves for a shallow water equation: a geometric approach
    • Constantin A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50 2 (2000) 321-362
    • (2000) Ann. Inst. Fourier (Grenoble) , vol.50 , Issue.2 , pp. 321-362
    • Constantin, A.1
  • 10
    • 60349102715 scopus 로고    scopus 로고
    • The hydro-dynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
    • Constantin A., and Lannes D. The hydro-dynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 193 (2009) 165-186
    • (2009) Arch. Ration. Mech. Anal. , vol.193 , pp. 165-186
    • Constantin, A.1    Lannes, D.2
  • 11
    • 33846053142 scopus 로고    scopus 로고
    • Inverse scattering transform for the Camassa-Holm equation
    • Constantin A., Gerdjikov V.S., and Ivanov R.I. Inverse scattering transform for the Camassa-Holm equation. Inverse Problems 22 (2006) 2197-2207
    • (2006) Inverse Problems , vol.22 , pp. 2197-2207
    • Constantin, A.1    Gerdjikov, V.S.2    Ivanov, R.I.3
  • 12
  • 13
    • 35248855142 scopus 로고    scopus 로고
    • Particle trajectories in solitary water waves
    • Constantin A., and Escher J. Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. 44 (2007) 423-431
    • (2007) Bull. Amer. Math. Soc. , vol.44 , pp. 423-431
    • Constantin, A.1    Escher, J.2
  • 14
    • 0032374820 scopus 로고    scopus 로고
    • Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation
    • Constantin A., and Escher J. Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math. 51 (1998) 475-504
    • (1998) Comm. Pure Appl. Math. , vol.51 , pp. 475-504
    • Constantin, A.1    Escher, J.2
  • 15
    • 0242350978 scopus 로고    scopus 로고
    • Geodesic flow on the diffeomorphism group of the circle
    • Constantin A., and Kolev B. Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78 (2003) 787-804
    • (2003) Comment. Math. Helv. , vol.78 , pp. 787-804
    • Constantin, A.1    Kolev, B.2
  • 18
    • 0000380413 scopus 로고    scopus 로고
    • Stability of the Camassa-Holm solitons
    • Constantin A., and Strauss W. Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 12 (2002) 415-422
    • (2002) J. Nonlinear Sci. , vol.12 , pp. 415-422
    • Constantin, A.1    Strauss, W.2
  • 19
    • 0000985293 scopus 로고    scopus 로고
    • Wave breaking for nonlinear nonlocal shallow water equations
    • Constantin A., and Escher J. Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181 (1998) 229-243
    • (1998) Acta Math. , vol.181 , pp. 229-243
    • Constantin, A.1    Escher, J.2
  • 20
    • 0043159153 scopus 로고    scopus 로고
    • A note on well-posedness for Camassa-Holm equation
    • Danchin R. A note on well-posedness for Camassa-Holm equation. J. Differential Equations 192 2 (2003) 429-444
    • (2003) J. Differential Equations , vol.192 , Issue.2 , pp. 429-444
    • Danchin, R.1
  • 21
    • 28044471774 scopus 로고    scopus 로고
    • Time periodic solution of the viscous Camassa-Holm equation
    • Fu Y.P., and Guo B.L. Time periodic solution of the viscous Camassa-Holm equation. J. Math. Anal. Appl. 313 (2006) 311-321
    • (2006) J. Math. Anal. Appl. , vol.313 , pp. 311-321
    • Fu, Y.P.1    Guo, B.L.2
  • 22
    • 47949103476 scopus 로고    scopus 로고
    • Controllability and asymptotic stabilization of the Camassa-Holm equation
    • Glass O. Controllability and asymptotic stabilization of the Camassa-Holm equation. J. Differential Equations 245 (2008) 1584-1615
    • (2008) J. Differential Equations , vol.245 , pp. 1584-1615
    • Glass, O.1
  • 23
    • 29144482893 scopus 로고    scopus 로고
    • Two new types of bounded waves of CH-γ equations
    • Guo B.L., and Liu Z.R. Two new types of bounded waves of CH-γ equations. Sci. China Ser. A 48 (2005) 1618-1630
    • (2005) Sci. China Ser. A , vol.48 , pp. 1618-1630
    • Guo, B.L.1    Liu, Z.R.2
  • 24
    • 23444457784 scopus 로고    scopus 로고
    • Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation
    • Hakkaev S., and Kirchev K. Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation. Comm. Partial Differential Equations 30 (2005) 761-781
    • (2005) Comm. Partial Differential Equations , vol.30 , pp. 761-781
    • Hakkaev, S.1    Kirchev, K.2
  • 25
    • 0002405102 scopus 로고    scopus 로고
    • The Cauchy problem for an integrable shallow water equation
    • Himonas A., and Misiolek G. The Cauchy problem for an integrable shallow water equation. Differential Integral Equations 14 7 (2001) 821-831
    • (2001) Differential Integral Equations , vol.14 , Issue.7 , pp. 821-831
    • Himonas, A.1    Misiolek, G.2
  • 26
    • 0037171407 scopus 로고    scopus 로고
    • Camassa-Holm, Korteweg de Vries and related models for water waves
    • Johnson R.S. Camassa-Holm, Korteweg de Vries and related models for water waves. J. Fluid Mech. 455 (2002) 63-82
    • (2002) J. Fluid Mech. , vol.455 , pp. 63-82
    • Johnson, R.S.1
  • 27
    • 1342300742 scopus 로고    scopus 로고
    • On solutions of the Camassa-Holm equation
    • Johnson R.S. On solutions of the Camassa-Holm equation. Proc. R. Soc. Lond. 459 (2003) 1687-1708
    • (2003) Proc. R. Soc. Lond. , vol.459 , pp. 1687-1708
    • Johnson, R.S.1
  • 28
    • 84990617895 scopus 로고
    • Commutator estimates and the Euler and Navier-Stokes equations
    • Kato T., and Ponce G. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988) 891-907
    • (1988) Comm. Pure Appl. Math. , vol.41 , pp. 891-907
    • Kato, T.1    Ponce, G.2
  • 29
    • 0033483085 scopus 로고    scopus 로고
    • The Camassa-Holm equation as a geodesic flow on the diffeomorphism group
    • Kouranbaeva S. The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40 (1999) 857-868
    • (1999) J. Math. Phys. , vol.40 , pp. 857-868
    • Kouranbaeva, S.1
  • 30
    • 42949104678 scopus 로고    scopus 로고
    • The compact and noncompact structures for two types of generalized Camassa-Holm-KP equations
    • Lai S.Y., and Xu Y. The compact and noncompact structures for two types of generalized Camassa-Holm-KP equations. Math. Comput. Modelling 47 (2008) 1089-1098
    • (2008) Math. Comput. Modelling , vol.47 , pp. 1089-1098
    • Lai, S.Y.1    Xu, Y.2
  • 31
    • 26844435359 scopus 로고    scopus 로고
    • Traveling wave solutions of the Camassa-Holm equation
    • Lenells J. Traveling wave solutions of the Camassa-Holm equation. J. Differential Equations 217 (2005) 393-430
    • (2005) J. Differential Equations , vol.217 , pp. 393-430
    • Lenells, J.1
  • 32
    • 0034688728 scopus 로고    scopus 로고
    • Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation
    • Li Y., and Olver P. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162 (2000) 27-63
    • (2000) J. Differential Equations , vol.162 , pp. 27-63
    • Li, Y.1    Olver, P.2
  • 33
    • 0031989845 scopus 로고    scopus 로고
    • Shallow water equation as a geodesic flow on the Bott-Virasoro group
    • Misiolek G.A. Shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24 (1998) 203-208
    • (1998) J. Geom. Phys. , vol.24 , pp. 203-208
    • Misiolek, G.A.1
  • 34
    • 0037259541 scopus 로고    scopus 로고
    • Fredholm determinants and the Camassa-Holm hierarchy
    • McKean H.P. Fredholm determinants and the Camassa-Holm hierarchy. Comm. Pure Appl. Math. 56 5 (2003) 638-680
    • (2003) Comm. Pure Appl. Math. , vol.56 , Issue.5 , pp. 638-680
    • McKean, H.P.1
  • 35
    • 61849146290 scopus 로고    scopus 로고
    • Well-posedness of modified Camassa-Holm equations
    • Mclachlan R., and Zhang X.Y. Well-posedness of modified Camassa-Holm equations. J. Differential Equations 246 (2009) 3241-3259
    • (2009) J. Differential Equations , vol.246 , pp. 3241-3259
    • Mclachlan, R.1    Zhang, X.Y.2
  • 36
    • 49749095564 scopus 로고    scopus 로고
    • An integrable semi-discretization of Camassa-Holm equation and its determinant solution
    • Ohta Y., Maruno K.I., and Feng B.F. An integrable semi-discretization of Camassa-Holm equation and its determinant solution. J. Phys. A 41 (2008) 1-30
    • (2008) J. Phys. A , vol.41 , pp. 1-30
    • Ohta, Y.1    Maruno, K.I.2    Feng, B.F.3
  • 37
    • 63449102722 scopus 로고    scopus 로고
    • Well posedness for a higher order modified Camassa-Holm equation
    • Olson E.A. Well posedness for a higher order modified Camassa-Holm equation. J. Differential Equations 246 (2009) 4154-4172
    • (2009) J. Differential Equations , vol.246 , pp. 4154-4172
    • Olson, E.A.1
  • 38
    • 0035479874 scopus 로고    scopus 로고
    • On the Cauchy problem for the Camassa-Holm equation
    • Rodriguez-Blanco G. On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal. 46 (2001) 309-327
    • (2001) Nonlinear Anal. , vol.46 , pp. 309-327
    • Rodriguez-Blanco, G.1
  • 39
    • 0043132083 scopus 로고    scopus 로고
    • New peaked solitary wave solutions of the generalized Camassa-Holm equation
    • Tian L.X., and Song X.Y. New peaked solitary wave solutions of the generalized Camassa-Holm equation. Chaos Solitons Fractals 19 (2004) 621-637
    • (2004) Chaos Solitons Fractals , vol.19 , pp. 621-637
    • Tian, L.X.1    Song, X.Y.2
  • 42
    • 17644367140 scopus 로고    scopus 로고
    • A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions
    • Wazwaz A.M. A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions. Appl. Math. Comput. 165 (2005) 485-501
    • (2005) Appl. Math. Comput. , vol.165 , pp. 485-501
    • Wazwaz, A.M.1
  • 43
    • 64149097831 scopus 로고    scopus 로고
    • Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation
    • Wu S.Y., and Yin Z.Y. Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation. J. Differential Equations 246 (2009) 4309-4321
    • (2009) J. Differential Equations , vol.246 , pp. 4309-4321
    • Wu, S.Y.1    Yin, Z.Y.2
  • 44
    • 0034394008 scopus 로고    scopus 로고
    • On the weak solutions to a shallow water equation
    • Xin Z., and Zhang P. On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53 11 (2000) 1411-1433
    • (2000) Comm. Pure Appl. Math. , vol.53 , Issue.11 , pp. 1411-1433
    • Xin, Z.1    Zhang, P.2
  • 45
    • 0036913863 scopus 로고    scopus 로고
    • On the uniqueness and large time behavior of the weak solutions to a shallow water equation
    • Xin Z., and Zhang P. On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Comm. Partial Differential Equations 27 (2002) 1815-1844
    • (2002) Comm. Partial Differential Equations , vol.27 , pp. 1815-1844
    • Xin, Z.1    Zhang, P.2
  • 46
    • 2942650679 scopus 로고    scopus 로고
    • On the blow-up scenario for the generalized Camassa-Holm equation
    • Yin Z.Y. On the blow-up scenario for the generalized Camassa-Holm equation. Comm. Partial Differential Equations 29 (2004) 867-877
    • (2004) Comm. Partial Differential Equations , vol.29 , pp. 867-877
    • Yin, Z.Y.1
  • 47
    • 34548424025 scopus 로고    scopus 로고
    • On the local well-posedness of 3-D water wave problem with vorticity
    • Zhang P., and Zhang Z.F. On the local well-posedness of 3-D water wave problem with vorticity. Sci. China Ser. A 50 (2007) 1065-1077
    • (2007) Sci. China Ser. A , vol.50 , pp. 1065-1077
    • Zhang, P.1    Zhang, Z.F.2
  • 48
    • 43549121216 scopus 로고    scopus 로고
    • Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations
    • Zheng Y., and Lai S.Y. Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations. Phys. Lett. A 372 (2008) 4141-4143
    • (2008) Phys. Lett. A , vol.372 , pp. 4141-4143
    • Zheng, Y.1    Lai, S.Y.2


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