Abstract and Applied Analysis

Persistence Property and Estimate on Momentum Support for the Integrable Degasperis-Procesi Equation

Zhengguang Guo and Liangbing Jin

Full-text: Open access

Abstract

It is shown that a strong solution of the Degasperis-Procesi equation possesses persistence property in the sense that the solution with algebraically decaying initial data and its spatial derivative must retain this property. Moreover, we give estimates of measure for the momentum support.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 390132, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449998

Digital Object Identifier
doi:10.1155/2013/390132

Mathematical Reviews number (MathSciNet)
MR3126754

Zentralblatt MATH identifier
1293.35019

Citation

Guo, Zhengguang; Jin, Liangbing. Persistence Property and Estimate on Momentum Support for the Integrable Degasperis-Procesi Equation. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 390132, 7 pages. doi:10.1155/2013/390132. https://projecteuclid.org/euclid.aaa/1393449998


Export citation

References

  • A. Degasperis and M. Procesi, “Asymptotic integrability,” in Symmetry and Perturbation Theory, pp. 23–37, World Science Publisher, River Edge, NJ, USA, 1999.
  • H. R. Dullin, G. A. Gottwald, and D. D. Holm, “Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,” Fluid Dynamics Research, vol. 33, no. 1-2, pp. 73–95, 2003.
  • 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 B. Kolev, “On the geometric approach to the motion of inertial mechanical systems,” Journal of Physics A, vol. 35, no. 32, pp. R51–R79, 2002.
  • 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 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. Bressan and A. Constantin, “Global conservative solutions of the Camassa-Holm equation,” Archive for Rational Mechanics and Analysis, vol. 183, no. 2, pp. 215–239, 2007.
  • A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,” Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475–504, 1998.
  • A. Constantin and W. A. Strauss, “Stability of the Camassa-Holm solitons,” Journal of Nonlinear Science, vol. 12, no. 4, pp. 415–422, 2002.
  • Z. Guo, “Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa-Holm equation,” Journal of Mathematical Physics, vol. 49, no. 3, Article ID 033516, 2008.
  • A. A. Himonas, G. Misiołek, G. Ponce, and Y. Zhou, “Persistence properties and unique continuation of solutions of the Camassa-Holm equation,” Communications in Mathematical Physics, vol. 271, no. 2, pp. 511–522, 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.
  • H. P. McKean, “Breakdown of a shallow water equation,” The Asian Journal of Mathematics, vol. 2, no. 4, pp. 867–874, 1998.
  • H. P. McKean, “Breakdown of the Camassa-Holm equation,” Communications on Pure and Applied Mathematics, vol. 57, no. 3, pp. 416–418, 2004.
  • G. Misiołek, “Classical solutions of the periodic Camassa-Holm equation,” Geometric and Functional Analysis, vol. 12, no. 5, pp. 1080–1104, 2002.
  • S. Shkoller, “Geometry and curvature of diffeomorphism groups with ${H}^{1}$ metric and mean hydrodynamics,” Journal of Functional Analysis, vol. 160, no. 1, pp. 337–365, 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. 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.
  • A. Degasperis, D. D. Holm, and A. N. I. Hone, “A new integrable equation with peakon solutions,” Theoretical and Mathematical Physics, vol. 133, no. 2, pp. 1463–1474, 2002.
  • Z. Guo, “Some properties of solutions to the weakly dissipative Degasperis-Procesi equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4332–4344, 2009.
  • D. Henry, “Persistence properties for the Degasperis-Procesi equation,” Journal of Hyperbolic Differential Equations, vol. 5, no. 1, pp. 99–111, 2008.
  • Y. Zhou, “Blow-up phenomenon for the integrable Degasperis-Procesi equation,” Physics Letters. A, vol. 328, no. 2-3, pp. 157–162, 2004.
  • A. Constantin, “On the scattering problem for the Camassa-Holm equation,” Proceedings of the Royal Society London A, vol. 457, no. 2008, pp. 953–970, 2001.
  • G. M. Coclite and K. H. Karlsen, “On the well-posedness of the Degasperis-Procesi equation,” Journal of Functional Analysis, vol. 233, no. 1, pp. 60–91, 2006.
  • D. Henry, “Infinite propagation speed for the Degasperis-Procesi equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 755–759, 2005.
  • O. G. Mustafa, “A note on the Degasperis-Procesi equation,” Journal of Nonlinear Mathematical Physics, vol. 12, no. 1, pp. 10–14, 2005.
  • N. Kim, “Eigenvalues associated with the vortex patch in 2-D Euler equations,” Mathematische Annalen, vol. 330, no. 4, pp. 747–758, 2004.
  • S.-G. Kang and T.-M. Tang, “The support of the momentum density of the Camassa-Holm equation,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2128–2132, 2011.
  • Y. Liu and Z. Yin, “Global existence and blow-up phenomena for the Degasperis-Procesi equation,” Communications in Mathematical Physics, vol. 267, no. 3, pp. 801–820, 2006.