Abstract and Applied Analysis

The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation

Nan Li, Shaoyong Lai, Shuang Li, and Meng Wu

Full-text: Open access

Abstract

A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of the pseudoparabolic regularization technique, its local well posedness in Sobolev space H S ( R ) with s > 3 / 2 is established via a limiting procedure. Provided that the initial value u 0 satisfies the sign condition and u 0 H s ( R )   ( s > 3 / 2 ) , it is shown that there exists a uniqueglobal solution for the equation in space C ( [ 0 , ) ; H s ( R ) ) C 1 ( [ 0 , ) ; H s 1 ( R ) ) .

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 532369, 26 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174041

Digital Object Identifier
doi:10.1155/2012/532369

Mathematical Reviews number (MathSciNet)
MR2910721

Zentralblatt MATH identifier
1237.35139

Citation

Li, Nan; Lai, Shaoyong; Li, Shuang; Wu, Meng. The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 532369, 26 pages. doi:10.1155/2012/532369. https://projecteuclid.org/euclid.aaa/1365174041


Export citation

References

  • R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.
  • A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 165–186, 2009.
  • R. S. Johnson, “Camassa-Holm, Korteweg-de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, no. 1, pp. 63–82, 2002.
  • D. Ionescu-Kruse, “Variational derivation of the Camassa-Holm shallow water equation,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 3, pp. 303–312, 2007.
  • R. S. Johnson, “The Camassa-Holm equation for water waves moving over a shear flow,” Japan Society of Fluid Mechanics. Fluid Dynamics Research. An International Journal, vol. 33, no. 1-2, pp. 97–111, 2003.
  • H.-H. Dai, “Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,” Acta Mechanica, vol. 127, no. 1–4, pp. 193–207, 1998.
  • A. Constantin and W. A. Strauss, “Stability of a class of solitary waves in compressible elastic rods,” Physics Letters. A, vol. 270, no. 3-4, pp. 140–148, 2000.
  • M. Lakshmanan, “Integrable nonlinear wave equations and possible connections to tsunami dynamics,” in Tsunami and Nonlinear Waves, A. Kundu, Ed., pp. 31–49, Springer, Berlin, Germany, 2007.
  • A. Constantin and R. S. Johnson, “Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,” Fluid Dynamics Research, vol. 40, no. 3, pp. 175–211, 2008.
  • A. Constantin, “The trajectories of particles in Stokes waves,” Inventiones Mathematicae, vol. 166, no. 3, pp. 523–535, 2006.
  • A. Constantin and J. Escher, “Particle trajectories in solitary water waves,” Bulletin of American Mathematical Society, vol. 44, no. 3, pp. 423–431, 2007.
  • A. Constantin and W. A. Strauss, “Stability of peakons,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp. 603–610, 2000.
  • A. Constantin and J. Escher, “Analyticity of periodic traveling free surface water waves with vorticity,” Annals of Mathematics, vol. 173, no. 1, pp. 559–568, 2011.
  • K. El Dika and L. Molinet, “Stability of multipeakons,” Annales de l'Institut Henri Poincare. Analyse Non Lineaire, vol. 26, no. 4, pp. 1517–1532, 2009.
  • A. Constantin and W. A. Strauss, “Stability of the Camassa-Holm solitons,” Journal of Nonlinear Science, vol. 12, no. 4, pp. 415–422, 2002.
  • A. Constantin and H. P. McKean, “A shallow water equation on the circle,” Communications on Pure and Applied Mathematics, vol. 52, no. 8, pp. 949–982, 1999.
  • A. Constantin, “On the inverse spectral problem for the Camassa-Holm equation,” Journal of Functional Analysis, vol. 155, no. 2, pp. 352–363, 1998.
  • A. Constantin, V. S. Gerdjikov, and R. I. Ivanov, “Inverse scattering transform for the Camassa-Holm equation,” Inverse Problems, vol. 22, no. 6, pp. 2197–2207, 2006.
  • H. P. McKean, “Integrable systems and algebraic curves,” in Global Analysis, vol. 755 of Lecture Notes in Mathematics, pp. 83–200, Springer, Berlin, Germany, 1979.
  • A. Constantin, T. Kappeler, B. Kolev, and P. Topalov, “On geodesic exponential maps of the Virasoro group,” Annals of Global Analysis and Geometry, vol. 31, no. 2, pp. 155–180, 2007.
  • G. Misiolek, “A shallow water equation as a geodesic flow on the Bott-Virasoro group,” Journal of Geometry and Physics, vol. 24, no. 3, pp. 203–208, 1998.
  • Z. Xin and P. Zhang, “On the weak solutions to a shallow water equation,” Communications on Pure and Applied Mathematics, vol. 53, no. 11, pp. 1411–1433, 2000.
  • Z. Xin and P. Zhang, “On the uniqueness and large time behavior of the weak solutions to a shallow water equation,” Communications in Partial Differential Equations, vol. 27, no. 9-10, pp. 1815–1844, 2002.
  • G. M. Coclite, H. Holden, and K. H. Karlsen, “Global weak solutions to a generalized hyperelastic-rod wave equation,” SIAM Journal on Mathematical Analysis, vol. 37, no. 4, pp. 1044–1069, 2005.
  • A. Bressan and A. Constantin, “Global conservative solutions of the Camassa-Holm equation,” Ar-chive for Rational Mechanics and Analysis, vol. 183, no. 2, pp. 215–239, 2007.
  • A. Bressan and A. Constantin, “Global dissipative solutions of the Camassa-Holm equation,” Analysis and Applications, vol. 5, no. 1, pp. 1–27, 2007.
  • H. Holden and X. Raynaud, “Dissipative solutions for the Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 24, no. 4, pp. 1047–1112, 2009.
  • H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation-a Lagran-gian point of view,” Communications in Partial Differential Equations, vol. 32, no. 10–12, pp. 1511–1549, 2007.
  • Y. A. Li and P. J. Olver, “Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,” Journal of Differential Equations, vol. 162, no. 1, pp. 27–63, 2000.
  • A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.
  • R. Beals, D. H. Sattinger, and J. Szmigielski, “Multi-peakons and a theorem of Stieltjes,” Inverse Problems, vol. 15, no. 1, pp. L1–L4, 1999.
  • D. Henry, “Persistence properties for a family of nonlinear partial differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 4, pp. 1565–1573, 2009.
  • S. Hakkaev and K. Kirchev, “Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 30, no. 4–6, pp. 761–781, 2005.
  • G. Rodríguez-Blanco, “On the Cauchy problem for the Camassa-Holm equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 46, no. 3, pp. 309–327, 2001.
  • Z. Yin, “On the blow-up scenario for the generalized Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 867–877, 2004.
  • Z. Yin, “On the Cauchy problem for an integrable equation with peakon solutions,” Illinois Journal of Mathematics, vol. 47, no. 3, pp. 649–666, 2003.
  • Y. Zhou, “Wave breaking for a periodic shallow water equation,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 591–604, 2004.
  • S. Lai and Y. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations, vol. 248, no. 8, pp. 2038–2063, 2010.
  • S. Y. Wu and Z. Yin, “Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4309–4321, 2009.
  • T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, vol. 448 of Lecture notes in Mathematics, pp. 25–70, Springer, Berlin, 1975.
  • T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988.
  • W. Walter, Differential and Integral Inequalities, Springer-Verlag, New York, NY, USA, 1970.
  • B. Kolev, “Lie groups and mechanics: an introduction,” Journal of Nonlinear Mathematical Physics, vol. 11, no. 4, pp. 480–498, 2004.
  • A. Constantin, “Existence of permanent and breaking waves for a shallow water equation: a geometric approach,” Annales de l'Institut Fourier, vol. 50, no. 2, pp. 321–362, 2000.
  • B. Kolev, “Poisson brackets in hydrodynamics,” Discrete and Continuous Dynamical Systems, vol. 19, no. 3, pp. 555–574, 2007.