广义的Camassa-Holm方程的弱适定性

Weak Well-posedness for the Generalized Camassa-Holm Equation

本文主要研究广义的Camassa-Holm方程Cauchy问题当初值u 0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。

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.