Abstract and Applied Analysis

The Local Strong and Weak Solutions for a Generalized Novikov Equation

Meng Wu and Yue Zhong

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Abstract

The Kato theorem for abstract differential equations is applied to establish the local well-posedness of the strong solution for a nonlinear generalized Novikov equation in space C ( [ 0 , T ) , H s ( R ) ) C 1 ( [ 0 , T ) , H s - 1 ( R ) ) with s > ( 3 / 2 ) . The existence of weak solutions for the equation in lower-order Sobolev space H s ( R ) with 1 s ( 3 / 2 ) is acquired.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 158126, 14 pages.

Dates
First available in Project Euclid: 4 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365099919

Digital Object Identifier
doi:10.1155/2012/158126

Mathematical Reviews number (MathSciNet)
MR2903806

Zentralblatt MATH identifier
1243.35044

Citation

Wu, Meng; Zhong, Yue. The Local Strong and Weak Solutions for a Generalized Novikov Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 158126, 14 pages. doi:10.1155/2012/158126. https://projecteuclid.org/euclid.aaa/1365099919


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References

  • V. Novikov, “Generalizations of the Camassa-Holm equation,” Journal of Physics A, vol. 42, no. 34, pp. 342002–342014, 2009.
  • A. N. W. Hone and J. P. Wang, “Integrable peakon equations with cubic nonlinearity,” Journal of Physics A, vol. 41, no. 37, pp. 372002–372010, 2008.
  • A. N. W. Hone, H. Lundmark, and J. Szmigielski, “Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation,” Dynamics of Partial Differential Equations, vol. 6, no. 3, pp. 253–289, 2009.
  • 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.
  • A. Constantin and J. Escher, “Particle trajectories in solitary water waves,” Bulletin of the American Mathematical Society, vol. 44, no. 3, pp. 423–431, 2007.
  • Z. Guo and Y. Zhou, “Wave breaking and persistence properties for the dispersive rod equation,” SIAM Journal on Mathematical Analysis, vol. 40, no. 6, pp. 2567–2580, 2009.
  • 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.
  • 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.
  • Y. Zhou, “Blow-up of solutions to the DGH equation,” Journal of Functional Analysis, vol. 250, no. 1, pp. 227–248, 2007.
  • Y. Zhou, “Blow-up of solutions to a nonlinear dispersive rod equation,” Calculus of Variations and Par-tial Differential Equations, vol. 25, no. 1, pp. 63–77, 2006.
  • L. Ni and Y. Zhou, “Well-posedness and persistence properties for the Novikov equation,” Journal of Differential Equations, vol. 250, no. 7, pp. 3002–3021, 2011.
  • F. Tiglay, “The periodic Cauchy problem for Novikov's equation,” http://arxiv.org/abs/1009.1820/.
  • S. 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, Lecture notes in Mathematics, Springer, Berlin, Germany, 1975.
  • T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Com-munications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988.