## Abstract and Applied Analysis

### A New Blow-Up Criterion for the DGH Equation

#### Abstract

We investigate the DGH equation. Analogous to the Camassa-Holm equation, this equation possesses the blow-up phenomenon. We establish a new blow up criterion on the initial data to guarantee the formulation of singularities in finite time.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 515948, 10 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495725

Digital Object Identifier
doi:10.1155/2012/515948

Mathematical Reviews number (MathSciNet)
MR2926884

Zentralblatt MATH identifier
1242.35198

#### Citation

Zhu, Mingxuan; Jin, Liangbing; Jiang, Zaihong. A New Blow-Up Criterion for the DGH Equation. Abstr. Appl. Anal. 2012 (2012), Article ID 515948, 10 pages. doi:10.1155/2012/515948. https://projecteuclid.org/euclid.aaa/1355495725

#### References

• R. Dullin, G. Gottwald, and D. Holm, “An integrable shallow water equation with linear and nonlinear dispersion,” Physical Review Letters, vol. 87, pp. 4501–4504, 2001.
• L. Tian, G. Gui, and Y. Liu, “On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation,” Communications in Mathematical Physics, vol. 257, no. 3, pp. 667–701, 2005.
• T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, vol. 4487 of Lecture Notes in Mathematics, pp. 25–70, Springer, Berlin, Germany, 1975.
• Y. Zhou, “Blow-up of solutions to the DGH equation,” Journal of Functional Analysis, vol. 250, no. 1, pp. 227–248, 2007.
• 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.
• B. Fuchssteiner and A. S. Fokas, “Symplectic structures, their Bäcklund transformations and hereditary symmetries,” Physica D, vol. 4, no. 1, pp. 47–66, 1981.
• R. S. Johnson, “Camassa-Holm, Korteweg-de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, pp. 63–82, 2002.
• 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.
• 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.
• 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.
• 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.
• H. P. McKean, “Breakdown of a shallow water equation,” Asian Journal of Mathematics, vol. 2, no. 4, pp. 867–874, 1998.
• Y. Zhou, “Wave breaking for a shallow water equation,” Nonlinear Analysis, vol. 57, no. 1, pp. 137–152, 2004.
• Y. Zhou, “Wave breaking for a periodic shallow water equation,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 591–604, 2004.
• Z. Jiang, L. Ni, and Y. Zhou, “Wave breaking of the Camassa-Holm equation,” Journal of Nonlinear Science, vol. 22, no. 1, pp. 235–245, 2012.
• Z. Jiang and S. Hakkaev, “Wave breaking and propagation speed for a class of one-dimensional shallow water equations,” Abstract and Applied Analysis, vol. 2011, Article ID 647368, 15 pages, 2011.
• Z. Guo and M. Zhu, “Wave breaking for a modified two-component Camassa-Holm system,” Journal of Differential Equations, vol. 252, pp. 2759–2770, 2012.
• 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. Bressan and A. Constantin, “Global dissipative solutions of the Camassa-Holm equation,” Analysis and Applications, vol. 5, no. 1, pp. 1–27, 2007.
• J. F. Toland, “Stokes waves,” Topological Methods in Nonlinear Analysis, vol. 7, no. 1, pp. 1–48, 1996.
• 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.
• 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.
• 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, “Finite propagation speed for the Camassa-Holm equation,” Journal of Mathematical Physics, vol. 46, no. 2, Article ID 023506, 4 pages, 2005.
• D. Henry, “Compactly supported solutions of the Camassa-Holm equation,” Journal of Nonlinear Mathematical Physics, vol. 12, no. 3, pp. 342–347, 2005.
• D. Henry, “Persistence properties for a family of nonlinear partial differential equations,” Nonlinear Analysis, vol. 70, no. 4, pp. 1565–1573, 2009.
• 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.
• L. Ni and Y. Zhou, “A new asymptotic behavior of solutions to the Camassa-Holm equation,” Proceedings of the American Mathematical Society, vol. 140, no. 2, pp. 607–614, 2012.
• A. Constantin, “The trajectories of particles in Stokes waves,” Inventiones Mathematicae, vol. 166, no. 3, pp. 523–535, 2006.
• A. Constantin, “Existence of permanent and breaking waves for a shallow water equation: a geometric approach,” Université de Grenoble. Annales de l'Institut Fourier, vol. 50, no. 2, pp. 321–362, 2000.
• B. Kolev, “Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations,” Philosophical Transactions of the Royal Society of London. Series A, vol. 365, no. 1858, pp. 2333–2357, 2007.
• Y. Zhou, “On solutions to the Holm-Staley $b$-family of equations,” Nonlinearity, vol. 23, no. 2, pp. 369–381, 2010.