Global Well-posedness Of 3-D Camassa-Holm Equations With Viscosity Terms

In this paper,we study 3-D Camassa-Holm equations with viscosity terms,which are closely related with many important physical situations.The 3-D Camassa-Holm equations are given as follows:(?)tm+u·?m+?uT·m+mdivu=0,(t,x)?R+×R3,Or in component formin which m=(I-?)u,u=u(t,x)=(u1,u2,u3),m=m(t,x).The corresponding 3-D Camassa-Holm equations with viscosity terms are given as(?)tm+u·?m+VuT·m+mdivu=v?m,wherev ?(0,1].At first,we construct the smooth approximation solutions of 3-D Camassa-Holm equations with given viscosity terms,by studying the uniform estimates,we derive the local well-posedness.Then through studying blow-up criteria,the local solutions can be extended to global situation.The article is separated into three parts,which is arranged from introduction to some estimates of heat equations to the global well-posedness of 3-D Camassa-Holm equations with viscosity terms.In chapter one,the research background,contents and main results are given,as well as the content organizations.The second chapter is the estimates in large time behavior and the proof of fundamental estimates.In chapter three,by using some main lemmas,classical Picard iteration theorem and Aubin-Lions lemma to show the local-posedness of viscosity equations.By using continuity criterion,the local solutions can be extended to the global situation.At last,the existence of the global solutions are proved.