高阶修正Camassa-Holm方程的Cauchy问题

Cauchy Problem of Higher-order Modified Camassa-Holm Equation

主要证明一类高阶修正的Camassa-Holm方程拥有哈密顿结构和建立在H2(R)适定性结果.首先证明高阶修正的Camassa-Holm方程拥有两个重要的守恒律.然后利用这两个重要的守恒律证明高阶修正的Camassa-Holm方程拥有哈密顿结构.并且使用Kato理论,证明高阶修正的Camassa-Holm方程在Hs(R)(s>3/2)中是局部适定的;利用两个重要的守恒律得到了一个重要的先验估计.结合局部适定性结果以及先验估计,对于初值u0∈H2(R),证明高阶修正的Camassa-Holm方程在H2(R)中是整体适定的.

In this paper, we mainly prove that the higher-order modified Camassa-Holm equation possesses the Hamitonian structure and establishes the global well-posedness result in H^2（R）. Firstly, it is shown that the higher-order modified Camassa-Holm equation possesses two important conseravtion laws. Then, we prove that the higher-order modified Camassa- Holm equation possesses the Hamitonian structure with the aid of two important conservation laws. By using Kato＇s theory, we prove that the Cauehy problem for the higher-order modified Camassa-Holm equation is locally well-posed for the initial data in H^s（R）（s〉3/2）. By using the two important conserved laws, we derive a prior estimate. Combining the prior estimate with the local well-posedness result, we derive the global well-posendess result of the Cauchy problem for the higher-order modified Camassa-Holm equation is globally well-posed for the initial data in H^2 （R）.