Studies On The Cauchy Problems Of Two Camassa-Holm Type Systems

In this thesis,we study the Cauchy problems of two CH type systems,which are very important in integrable systems.In chapter 1,we give a brief background and research status of shallow water equations,and the topics which will be considered in the thesis.In chapter 2,we recall the definition of nonhomogeneous Besov space,which is based on the Littlewood-Paley unit decomposition.Then,by introducing the Bony paraproduct decomposition and Bernstein inequalities,we offer the Moser-type estimates and transport estimate theory in Besov space,which play important roles in the next chapters.In chapter 3,we consider the Cauchy problem for a three-component Novikov system.Firstly,by using transport estimate theory,we prove the local well-posedness in(Bp,rs)3 with s>max{1/2,1/p},and the critical Besov space(B2,11/2)3 Then we give blow-up criteria of strong solutions in Sobolev space(Hs)3 when s>1/2.Finally,we consider the global existence and uniqueness of solutions for Geng-Xue system.We prove the strong solutions are global in(Hs)2 whit s>1/2,and by viscous approximate method,we prove the weak solutions are global in(L1∩L2)2.In chapter 4,we consider the properties of solutions for the three-component Novikov system.First we prove the persistence of solutions,which means if initial data is infinitesimal quantities of the e-α|x| as |x|→∞,then such property persists for the corresponding solutions with different decay rate.Finally,we derive a crucial lemma for traveling wave solution to show its x-asymmetrical property.And as a conclusion,we obtain the unique traveling wave solution for Geng-Xue system is x-symmetrical.Chapter 5 is devoted to expand the μmCH equation into multi-component cases.By extending the matrix of spectral problem of μmCH equation and take some suitable reductions,we obtain three two-component μmCH systems.Both systems admit bi-Hamiltonian structures.As an example,we consider the Cauchy problem of one 2μmCH system.We claim it is local well-posed in Hs(S)with s>1/2.Then,we offer a blow-up criterion of strong solutions in Sobolev space.Finally,with a direct computation,we illustrate that these systems also admit periodic peakons.