Non-uniform Continous Dependence For The High-dimensional Camassa-Holm Equations In Sobolev Spaces

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 u?,n which satsfying(?)tu?,n+u?,n·?u?,n=P(u?,n,u?,n)+E?,n+F?,n.Then,we take the difference between the approximate solution and the actual solution to get v:=v?,n=u?,n-u?,n which satisfying(?)tv+u?,n·?v+v·?u?,n=P(u?,n,v)+P(v,u?,n)-E?,n-F?,n.We can deduce the Sobolev norm of v?,n is infinitesimal sequence.Thus,we can transform the non-uniform continuous dependence of the actual solution u?,n into the non-uniform continuous dependence of the approximate solution u?,n.Then,according to the properties of u?,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.