Upper Semi-continuity Of Attractors For The Dissipative Camassa-Holm Equation With Additive Part

The main idea of this paper is to discuss the properties of solutions of the stochastic dissipative Camassa-Holm equation, and then prove the existence and semi-continuity of attractors by the solution of the equation in H01. First consider the stochastic dissipative Camassa-Holm equation: Among above, ε∈ G (0,1], λ> 0. W(t)= (W1(t),W2(t),…,Wm(t)) is a two-sided real-valued Wiener-process that defined on a complete probability space (Ω,F, P). Ω={ε∈C (R, Rm)ω(0)= 0}. P is a wiener measurement. F is a Borel-sigma algebra that induced by a compact open topology of Ω. (Ω,F, P,θt) is a distance dynamical systems. αi∈R,φi=φi(x)∈D(A), x∈I, i= 1,2, …, m.The first chapter mainly introduces the definition of dynamical system and attractors. And then discuss the background and research results of the Camassa-Holm equation. Beside this, some basic theory are given.In the second chapter, we study the stochastic dissipative Camassa-Holm e-quation in detail. Then prove the existence and uniqueness of the solution of the equation. And then describe the dynamical system of the solution.In the third chapter, according to a priori estimate, we prove the existence of absorbing set and asymptotic compactness of φε in H01. Finally prove the existence of attractors in H01.The forth chapter gives the result. According to prove the convergence of dynamical systems by the solution of the equation, the boundedness of absorbing set, and the compactness of attractors, we prove the upper semi-continuity of attractors for the stochastic Camassa-Holm equation in H01.