广东工业大学学报 ›› 2020, Vol. 37 ›› Issue (02): 80-86.doi: 10.12052/gdutxb.190053

• 综合研究 • 上一篇    下一篇

广义的Camassa-Holm方程的弱适定性

陈荣宁, 卫雪梅   

  1. 广东工业大学 应用数学学院, 广东 广州 510520
  • 收稿日期:2019-04-10 出版日期:2020-03-10 发布日期:2020-02-12
  • 通信作者: 卫雪梅(1972-),女,教授,主要研究方向为偏微分方程,E-mail:wxm_gdut@163.com E-mail:wxm_gdut@163.com
  • 作者简介:陈荣宁(1996-),男,硕士研究生,主要研究方向为偏微分方程
  • 基金资助:
    国家自然科学基金资助项目（11101095）；广东省高校特色创新类项目（2016KTSCX028）

Weak Well-posedness for the Generalized Camassa-Holm Equation

Chen Rong-ning, Wei Xue-mei   

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510520, China
  • Received:2019-04-10 Online:2020-03-10 Published:2020-02-12

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摘要： 本文主要研究广义的Camassa-Holm方程Cauchy问题当初值u0在空间H1（R）∩W1，∞（R）时解的弱适定性。首先运用特征线把广义的Camassa-Holm方程转化成类似常微分方程（Ordinary Differential Equation，ODE）的方程。其次运用ODE理论证明新方程解的局部存在唯一性。最后利用新方程与原方程的关系，证明原方程解的局部存在唯一性并且给出解对初值的弱连续依赖性。

关键词: 广义的Camassa-Holm方程, 特征线, 弱连续依赖性

Abstract: The weak well-posedness for the generalized Camassa-Holm equation with initial data u0 ∈ H1(R)∩W1,∞(R) is mainly considered. First, by introducing the characteristics, the generalized Camassa-Holm equation is transformed into an ODE(Ordinary Differential Equation)-similar equation. Then applying the ODE theory, the local existence and uniqueness of the solution to the new equation are proved. Finally, by using the relationship between the new equation and the original equation, the local existence and uniqueness of the solution are investigated, deriving the conclusion of the weak continuous dependence on initial data for the original equation.

Key words: Generalized Camassa-Holm equation, characteristics, weak continuous dependence

中图分类号: 

  • O175
[1] GRAYSHAN K, HIMONAS A A. Equations with peakon traveling wave solutions[J]. Advances in Dynamical Systems and Applications, 2013, 8(2):217-232
[2] GUAN C X, WANG J M, MENG Y P. Weak well-posedness for a modified two-component Camassa-Holm system[J]. Nonlinear Analysis, 2019, 178:247-265
[3] GUAN C X, ZHU H. Well-posedness of a class of solutions to an integrable two-component Camassa-Holm system[J]. Journal of Mathematical Analysis and Applications, 2019, 470:413-433
[4] WEI L, QIAO Z J, WANG Y, ZHOU S M. Conserved quantities, Global existence and blow-up for a generalized CH equation[J]. Discrete and Continuous Dynamical Systems, 2017, 37(3):1733-1748
[5] TU X, YIN Z Y. Global weak solutions for a generalized Camassa-Holm equation[J]. Mathematische Nachrichten, 2018, 291(16):2457-2475
[6] GUAN C X. Uniqueness of global conservative weak solutions for the modified two-component Camassa-Holm system[J]. Journal of Evolution Equations, 2018, 18(2):1003-1024
[7] GUAN C X, KARLSEN K H, YIN Z Y. Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation[J]. Contemporary Mathematics, 2010, 526:199-220
[8] GUAN C X, YAN K, WEI X M. Lipschitz metric for the modified two-component Camassa-Holm system[J]. Analysis and Applications, 2018, 16(2):159-182
[9] GUAN C X, YIN Z Y. Global weak solutions for a modified two-component Camassa-Holm equation[J]. Annales de I'Institut Henri poincaré/Analyse Non Linéaire, 2011, 28(4):623-641
[10] GUAN C X, YIN Z Y. On the global weak solutions for a modified two-component Camassa-Holm equation[J]. Mathematische Nachrichten, 2013, 286(13):1287-1304
[11] BAROSTICHI R F, HIMONAS A A, PETRONILHO G. Global analyticity for a generalized Camassa-Holm equation and decay of the radius of spatical analyticity[J]. Journal of Differential Equations, 2017, 263(1):732-764
[12] HIMONAS A A, HOLLIMAN C. The Cauchy problem for a generalized Camassa-Holm equation[J]. Advances in Differential Equations, 2014, 19(1-2):161-200
[13] YAN W, LI Y S. The Cauchy problem for the modified two-component Camassa-Holm system in critical Besov Space[J]. Annales de I'Institut Henri poincaré/Analyse Non Linéaire, 2015, 32:443-469
[14] ZHOU S M. The Cauchy problem for a generalized b-equation with higher-order nonlinearities in critical besov spaces and weighted spaces[J]. Discrete and Continuous Dynamical Systems, 2014, 3-4(11):4967-4986
[15] LINARES F, PONCE G, SIDERIS T C. Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons[EB/OL]. (2016-12-2)[2019-04-07]. https://arxiv.org/abs/1609.06212.
[16] CAMASSA R, HOLM D D. An integrable shallow water equation with peaked solutions[J]. Physical Review Letters, 1993, 71(11):1661-1664
[17] GIUSEPPE G,RAFFAELE V,SEBASTIAN W. Symmetry and Perturbation Theory[M]. Singapore:World Scientific, 1999:23-37.
[18] NOVIKOV V. Generalizations of the Camassa-Holm equation[J]. Journal of Physics A:Mathematical and Theoretical, 2009, 42(34):2002-2015
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