一类五阶Camassa-Holm方程的Blow-Up准则
Blow-Up Criteria for a Fifth-Order Camassa-Holm Equation
DOI: 10.12677/AAM.2021.105175, PDF,   
作者: 卢宁宁：华北电力大学数理学院，北京
关键词: 五阶Camassa-Holm方程；Blow-Up准则；Fifth-Order Camassa-Holm Equation； Blow-Up Criteria
摘要: Camassa-Holm方程作为一个新的完全可积的色散浅水波方程，在过去的几十年内受到了广泛关注。本文中我们主要研究的是一类五阶Camassa-Holm方程的Blow-up准则。在三种不同情况下，我们通过对方程的解作估计，得到了这类方程的解在有限时间内爆破的一个必要条件。因此，本文的研究结果丰富了广义的Camassa-Holm方程的爆破性质。
Abstract: As a new completely integrable dispersive shallow-water wave equation, Camassa-Holm equation has been widely concerned in the past few decades. In this paper, we mainly study the blow-up criteria for a fifth-order Camassa-Holm equation. By estimating the solution of the equation in three different cases, we obtain a necessary condition for the solution of the equation to blow-up in finite time. Therefore, the results of this paper enrich the blow-up properties of the generalized Camassa-Holm equation.
文章引用：卢宁宁. 一类五阶Camassa-Holm方程的Blow-Up准则[J]. 应用数学进展, 2021, 10(5): 1647-1653. https://doi.org/10.12677/AAM.2021.105175

参考文献

[1] Camassa, R. and Holm, D.D. (1993) An Integrable Shallow Water Equation with Peaked Solitons. Physical Review Letters, 71, 1661-1664. [Google Scholar] [CrossRef] 
[2] Constantin, A. and Escher, J. (2007) Particle Trajectories in Solitary Water Waves. Bulletin of the American Mathematical Society, 44, 423-431. [Google Scholar] [CrossRef] 
[3] Chen, A., Deng, T. and Qiao, Z. (2020) Stability of Peakons and Periodic Peakons for a Nonlinear Quartic Camassa-Holm Equation.
[4] El Dika, K. and Molinet, L. (2009) Stability of Multipeakons. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 26, 1517-1532. [Google Scholar] [CrossRef] 
[5] Lenells, J. (2005) Traveling Wave Solutions of the Camassa-Holm Equation. Journal of Differential Equations, 217, 393-430. [Google Scholar] [CrossRef] 
[6] Gui, G., Yue, L. and Olver, P.J. (2013) Wave-Breaking and Peakons for a Modified Camassa-Holm Equation. Communications in Mathematical Physics, 319, 731-759. [Google Scholar] [CrossRef] 
[7] Gui, G., Liu, Y. and Luo, T. (2018) Model Equations and Traveling Wave Solutions for Shallow-Water Waves with the Coriolis Effect. Journal of Nonlinear Science, 29, 993-1039. [Google Scholar] [CrossRef] 
[8] Constantin, A. and Escher, J. (1998) Wave Breaking for Nonlinear Nonlocal Shallow Water Equations. Acta Mathematica, 181, 229-243. [Google Scholar] [CrossRef] 
[9] Constantin, A. and Escher, J. (2000) On the Blow-Up Rate and the Blow-Up Set of Breaking Waves for a Shallow Water Equation. Mathematische Zeitschrift, 233, 75-91. [Google Scholar] [CrossRef] 
[10] Constantin, A. (2000) Existence of Permanent and Breaking Waves for a Shallow Water Equation: A Geometric Approach. Annales de l’Institut Fourier, 50, 321-362. [Google Scholar] [CrossRef] 
[11] Constantin, A. and Escher, J. (1998) Global Existence and Blow-Up for a Shallow Water Equation. Annali della Scuola normale superiore di Pisa, Classe di scienze, 26, 303-328.
[12] Mckean, H.P. (2010) Breakdown of the Camassa-Holm Equation. Communications on Pure and Applied Mathematics, 57, 416-418. [Google Scholar] [CrossRef] 
[13] Constantin, A. (2000) On the Blow-Up of Solutions of a Periodic Shallow Water Equation. Journal of Nonlinear Science, 10, 391-399. [Google Scholar] [CrossRef] 
[14] Gui, G. and Liu, Y. (2015) On the Global Existence and Wave-Breaking Criteria for the Two-Component Camassa-Holm System. Journal of Functional Analysis, 258, 4251-4278. [Google Scholar] [CrossRef] 
[15] Tian, L., Zhang, P. and Xia, L. (2011) Global Existence for the Higher-Order Camassa-Holm Shallow Water Equation. Nonlinear Analysis Theory Methods & Applications, 74, 2468-2474. [Google Scholar] [CrossRef] 

为你推荐