| In this paper,we study the non-uniform dependence for the solutions of the high-dimensional CamassaHolm equations in Sobolev spaces.Motivated by the Hilmonas’s paper[24,26],we use the structure of the high-dimensional Camassa-Holm equations and construct a stuitable approximate solution uω,n which satsfying(?)tuω,n+uω,n·▽uω,n=P(uω,n,uω,n)+Eω,n+Fω,n.Then,we take the difference between the approximate solution and the actual solution to get v:=vω,n=uω,n-uω,n which satisfying(?)tv+uω,n·▽v+v·▽uω,n=P(uω,n,v)+P(v,uω,n)-Eω,n-Fω,n.We can deduce the Sobolev norm of vω,n is infinitesimal sequence.Thus,we can transform the non-uniform continuous dependence of the actual solution uω,n into the non-uniform continuous dependence of the approximate solution uω,n.Then,according to the properties of uω,n,we obtain our result.This paper is divided into two chapters which structure is as follow:The first chapter is devoted to give an introduction to the background,the usual notations,the basic concepts and the preparatory knowledge.The second chapter,we use some properties of the Littlewood-Paley decomposition and inhomogeneous Besov space to give the proof of the main theorem. |
