The Well-posedness And Blow-up Problems Of Fornberg-Whitham Equations

The Fornberg-Whitham equations are a class of nonlinear water wave models,whose solutions can describe the motion of water waves,such as propagating waves,solitons and wave breaking.They can be expressed as nonlinear nonlocal equations,where the nonlinear term can describe the characteristics of water waves and the nonlocal term reflects the dispersive properties of the equations.Based on the theory of partial differential equation,the thesis is devoted to studying the existence and qualitative properties of the Fornberg-Whitham equations.The structure of the thesis is as follows:The chapter 1 briefly introduces the research background,research status and main research work of this thesis.The chapter 2 studies the existence of the traveling wave solutions to the Fornberg-Whitham equations.Firstly,we use the minimization principle to prove the existence of solitary waves for the Fornberg-Whitham equation,where the Penalization function,the construction of global minimization sequences and concentration-compactness principle play an important role.Secondly,we establish the orbital stability and decay behavior of solitary wave solutions.Finally,we prove the existence of traveling wave solutions of the two-component Fornberg-Whitham system by using Crandall-Rabinowitz local bifurcation theorem.The chapter 3 considers the Cauchy problem of Fornberg-Whitham system.Firstly,the local well-posedness of the system in Hs(R)×Hs-1(R)(s>3/2)is proved by Galerkin approximation method.That is,the solutions of the system are unique and depends continuously on the initial data.Secondly,we prove that the well-posedness is sharp in the sense that the data-to-solution map is not uniformly continuous by the method of approximate solutions.The chapter 4 deals with the singularity problem of the Fornberg-Whitham system.We use the energy method to obtain the priori estimate and give the blow-up criterion of the system,which means that the solutions blow up in Hs(R)× Hs-1(R)(s>3/2)if and only if the wave breaking occurs.Besides,we also give the sufficient conditions for the occurrence of wave breaking.Based on blow-up criterion,we construct special initial data and prove the ill-posedness of the system in critical Sobolev space H3/2(R)× H1/2(R)by the norm inflaction.