Well-posedness And Blow-up For Some Generalized Shallow Water Wave Equations

In this paper,we mainly consider some kinds of generalized shallow water wave equations describing the fluid motion in shallow water environment.They are the gen-eralization of common shallow water wave equations in the integrable system,such as Camassa-Holm,Novikov,Fokas-Olver-Rosenau-Qiao equation and so on.They have sig-nificant background in water wave.We focus on the local well-posedness in Besov spaces,the blow-up criterion in finite time,the non-uniform dependence,and the persistence properties of strong solution.We first we study the Cauchy problem to the generalized Fokas-Olver-Resenau-Qiao equation in Chapter Two,mt+((u2-ux2)Qm)x=0 t>0,x?R.where m=u-uxx,Q? 1.As a generalization of Fokas-Olver-Resenau-Qiao equation,high nonlinear terms and derivative term "ux"result in many difficulties in the study of well-posedness.By means of transport equation and Littlewood-Paley theory and Osgood Lemma,we obtain the local well-posedness in Besov spaces Bp,rs(R)with s>max{2+1/p,5/2}and in critical space B2,15/2(R).Then two blow-up criterion are addressed by using particle line and commutator estimates.Here need explanation is that there is no any well-posedness or blow-up results about this equation,thus our work fills the research vacancy of this respect.Then in Chapter Three,We investigate the the higher dimensional Camassa-Holm equations mt+u·?m+?uT·m+m(div u)=0,t? 0,x?Td.with periodic boundary condition,where Td=(R/2?Z)d is the torus.Motivated by the issues of non-uniform dependence on initial data for the Camassa-Holm equation,combined with the research of the non-uniform dependence for the Euler equations in multi-dimensional case,We firstly introduce the approximate solutions,and estimate the error by using the Fourier series and transport theory.Thus,we show that when s>1+d/2(d?2)and 1 ?r??,the solution map is not uniformly continuous from B2,rs(Td)into E2,3s(T).Here T>0,and E2,rs(T)is defined as Our results is the supplement of the analysis for higher dimensional Camassa-Holm equa-tions,as well as generalizing the related results for Camassa-Holm equations.Last in Chapter Four,we consider the Cauchy problem of the following generalized two-component rotational b-family system Motivated by by the work of Brandolese on the Camassa-Holm equation in the weighted Sobolev spaces,we establish the persistence properties in weighted Lp(2 ?p ??)spaces for a large class of "moderate" weights.So far,it seems that there is no any persistence results this system.Our result is consistent the persistence property for Camassa-Holm equation,and other two-component Camassa-Holm equations,and can be applied to generalized two-component equations.In particular,by choosing different weighted functions,we can derive the asymptotic behavior from the persistence properties.Then we derive two blow-up results for the strong solutions to the system.In order to overcome the difficulty arising from no conservation law and higher order nonlinearity,we creatively take advantage of the intrinsic but special structure of the equation,perform an energy method on the system,and make use of commutator estimate.Then,we first figure out the estimates of ||u(t,·)||L2 and ||ux(t,·)||L2.Then we are able to provide the||u(t,·)||L? estimate||u(t,·)||L??eM1?(b-2)T/2(||u0||L2+||u0x||||L2+(1-2?A)||?0||L2),by using Gagliardo-Nirenberg inequality and thus complete the blow-up proof.Here,suppose there exists M1>0 such that ux(t,x)?-M1 for all(t,x)?[0,T)× R.