Low Regularity Theory On Some Nonlinear Dispersive Equations And Wave Equations

This doctoral dissertation is devoted to study some basic theories on the disper-sive equations, which is concerned with local well-posedness, global well-posedness, ill-posedness and scattering. It is made up of seven chapters.Chapter one is preface, which contains the introduction of the models studied in thisbook and the main methods used in this book. In Section 1, we starting with backgroundof the models, the previous works concerted with the models. Moreover, it contains thedescription of the main results obtained and the main innovations in this thesis. In Section2 and Section 3, we introduce the main methods used in this thesis, which are the"Fourierrestriction norm method"and the"I-method". Moreover, we present some preliminaryestimates in this chapter, such as Strichartz estimates and bilinear Strichartz estimates.In Chapter two, we study the local well-posedness, ill-posedness and global well-posedness of the Cauchy problem of the Schr¨odinger-Korteweg-de Vries equations. First,we show the the local well-posedness of this system by Bourgain's argument. We im-prove the previous works of Bekiranov, Ogawa, Ponce (1997) and Corcho, Linares (2007).Moreover, we obtain the best indices for which the bilinear estimates hold. Second, weprove the ill-posedness, which implies that they are sharp in some well-posedness thresh-olds. Particularly, we obtain the local well-posedness for the initial data belonging toH? 136+(R)×H?34+(R) in the resonant case and belonging to L2(R)×H?43+(R) in thenon-resonant case, it is almost the optimal except the endpoint. At last we establish theglobal well-posedness results in Hs(R)×Hs(R) when s > 12 no matter in the resonantcase or in the non-resonant case, which is achieved by I-method,with some special mul-tilinear estimates (this technique is rather general, and first developed by the author).This improves the result of Pecher (2005).In Chapter three, we first study the I-method with a further"partial refined"ar-gument which is introduced by My co-workers and I. Then we apply this argument tosome dispersive equations. As an example, we consider the Cauchy problem of the cubicnonlinear Schro¨dinger equation with derivative in Sobolev spaces Hs(R). This equationis known to ill-posedness for s < 21. In addition, Colliander, Keel, Sta?lani, Takaoka, Tao (I-term) (2002) obtained global well-posedness for s > 21 by I-method. While theproblem whether it is global well-posedness in the endpoint space H 21(R) remained open.In the second part of this Chapter, we study this open problem, and give the positiveanswer. In Section 3, as other applications of the"partial refined"argument, we studythe periodic and non-periodic mass-critical Schro¨dinger equation, we prove that in bothcase, the equations are globally well-posed in Hs(R) (or Hs(T)), for any s > 2/5, theresults improve the previous works of Bourgain's and I-term's.In chapter four, we study the global well-posedness for solutions to the defocusingcubic wave equation on three dimension. We show that the equation is globally well-posedin Hs(R3)×Hs?1(R3), for s > 0.7. The result obtained here improve the previous worksof Kenig, Ponce, Vega's (2000), Gallagher, Planchon's (2003), Bahouri, Chemin's (2006)and Roy's (2009). The improvement is mainly based on a better estimate on nonlinearterm.In chapter five, we study the global well-posedness and scattering for solutions to thedefocusing H 21-subcritical Hartree equation on three dimension. Our result improves theprevious one obtained by Miao, Xu and Zhao (2009). The main approaches are I-methodand the almost Morawetz estimates.In chapter six, we study the global well-posedness for solutions to the Benjaminequation and show that it is globally well-posed in Sobolev spaces Hs(R) for s > ?3/4.In this paper, we use some argument to obtain a good estimative for the lifetime of thelocal solution.In chapter seven, we first study a fifth-order KdV-type equation and study the bilinearestimate of nonlinear term of this equation. We find the bilinear estimate based on theusual Bourgain space behaves weak (indeed, it does not hold when the regularity index isbelow to 41). For this reason, we define a modified Bourgain space and obtain a bilinearestimate based on this. By this bilinear estimate, we give some better local result for thefifth-order KdV-type equation. At the end, by showing a ill-poseded result, we prove thatthe local result we obtained above is sharp.