几类非线性色散方程和波动方程的低正则性理论
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摘要
本文主要涉及非线性色散波方程的基本理论:局部适定性,不适定性,整体适定性以及散射理论.这些理论是在初值的正则性低于通常的质量空间或能量空间的意义下建立的.
本文分为七章,按照各类方程的研究成果划分.
第一章为绪论,本章涉及本课题研究的背景和研究进展,以及所用的基本方法和基本引理.本章分为三节,第一节我们表述在本课题中所涉及的模型的背景和研究进展,其中涉及了对本课题所获的主要结论和主要创新之处的描述.第二,三节我们介绍本文所涉及的基本方法,其中重点介绍Bourgain的Fourier限制模方法和I方法.同时,在这两节当中,我们也将介绍一些预备性估计,如Strichartz估计,双线性Strichartz估计等.
第二章主要研究Schr¨odinger-Korteweg-de Vries系统的局部适定性,整体适定性,和不适定性.首先,我们应用Bourgain的Fourier限制模方法建立该系统的局部适定性,这一结果改进了Bekiranov,Ogawa和Ponce(1997)以及Corcho和Linares (2007)等人的工作.并且我们获得了一些双线性估计的最佳指标.进一步,我们获得了该系统的不适定性结论.该结论表明我们之前所获得的局部适定性指标在某些区域上是最佳的.特别地,我们获得了共振系统的最佳下界指标(端点除外)L2(R)×H?43+(R),以及非共振系统的最佳下界指标H? 136+(R)×H?43+(R).最后,应用I方法,并结合一些特殊的多重线性估计(这一技术具有一定的一般性,首先由作者本人引入),我们建立了该系统的整体适定性.结论是:不论是共振还是非共振情形,该系统均在Hs(R)中,当s > 21时整体适定.该结果较大程度地改进了Pecher (2005)的工作.
第三章主要研究带部分修正技术的I方法在几类色散方程中的应用.本章前半部分我们详细介绍我和我的合作者们发展的,在I方法框架下的部分修正技术。这一技术在应用中卓有成效,本章的第二节我们以带库仑位势的导数Schr¨odinger方程为例,介绍它在色散方程整体适定性中的应用。导数Schr¨odinger方程在Sobolev空间Hs(R)中,当s < 21时是不适定的,这一结论被人们所熟知.同时,著名团队Colliander,Keel,Sta?lani,Takaoka 和Tao (I-term) 在2002年获得了该方程当指标s > 21时的整体适定性.关于临界空间H21(R)的整体适定性问题一直公开.本章第二节,我们应用带部分修正技术的I方法对这个问题给出了肯定地回答.本章第三节,我们进一步运用这一技术研究周期和非周期情形的质量临界的Schr¨odinger方程,并获得了方程在Hs(R)(或Hs(T)),s > 25时的整体适定性.这一结果改进了Bourgain(菲尔兹奖获得者)和I-term的结论。
第四章主要研究散焦情形的三维库仑位势的波动方法的整体适定性.本章中,我们证明:当初值属于Hs(R3)×Hs?1(R3),当s > 0.7时,方程的整体适定性.我们所获得的结果改进了Kenig,Ponce和Vega (2000),Gallagher和Planchon (2003),Bahouri和Chemin (2006)以及Roy(2009)的工作.这一改进主要来自于我们获得了一个更优的非线性光滑估计.
第五章,我们研究了三维散焦形H12次临界的Hartree方程.运用I方法,几乎Morawetz估计等技术,我们获得了方程低于能量空间的整体适定性和散射性质.这一结论改进了Miao,Xu和Zhao(2009)的工作.
第六章,我们研究Benjamin方程.这是一个不具有伸缩不变解的方程.基于I方法,这类方程的整体适定性主要有两个要素决定:第一,修正能量的误差估计;第二,局部生命跨度.在这一章中,我们证明了一种特殊的双线性估计(这一技术的思想来源于第二章Schr¨odinger-Korteweg-de Vries系统整体适定性的研究),从而延长局部生命跨度.最终我们证明了当初值属于Hs(R)中,当s > ?43时的整体适定性.
第七章,我们从一类五阶KdV方程的特点出发,研究这类方程所含非线性项的双线性估计.首先,我们揭示了该非线性项,基于通常Bourgain空间的双线性估计并不理想(指标s < 14时,不成立).为了进一步的研究,我们构建了修正的Bourgain空间,并证明了基于这类修正的Bourgain空间的双线性估计,从而证明了这类五阶KdV方程的局体适定性.最后,我们证明一个不适定性结论,从而揭示我们所获得的局部适定性结论是最优的.
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.
本文分为七章,按照各类方程的研究成果划分.
第一章为绪论,本章涉及本课题研究的背景和研究进展,以及所用的基本方法和基本引理.本章分为三节,第一节我们表述在本课题中所涉及的模型的背景和研究进展,其中涉及了对本课题所获的主要结论和主要创新之处的描述.第二,三节我们介绍本文所涉及的基本方法,其中重点介绍Bourgain的Fourier限制模方法和I方法.同时,在这两节当中,我们也将介绍一些预备性估计,如Strichartz估计,双线性Strichartz估计等.
第二章主要研究Schr¨odinger-Korteweg-de Vries系统的局部适定性,整体适定性,和不适定性.首先,我们应用Bourgain的Fourier限制模方法建立该系统的局部适定性,这一结果改进了Bekiranov,Ogawa和Ponce(1997)以及Corcho和Linares (2007)等人的工作.并且我们获得了一些双线性估计的最佳指标.进一步,我们获得了该系统的不适定性结论.该结论表明我们之前所获得的局部适定性指标在某些区域上是最佳的.特别地,我们获得了共振系统的最佳下界指标(端点除外)L2(R)×H?43+(R),以及非共振系统的最佳下界指标H? 136+(R)×H?43+(R).最后,应用I方法,并结合一些特殊的多重线性估计(这一技术具有一定的一般性,首先由作者本人引入),我们建立了该系统的整体适定性.结论是:不论是共振还是非共振情形,该系统均在Hs(R)中,当s > 21时整体适定.该结果较大程度地改进了Pecher (2005)的工作.
第三章主要研究带部分修正技术的I方法在几类色散方程中的应用.本章前半部分我们详细介绍我和我的合作者们发展的,在I方法框架下的部分修正技术。这一技术在应用中卓有成效,本章的第二节我们以带库仑位势的导数Schr¨odinger方程为例,介绍它在色散方程整体适定性中的应用。导数Schr¨odinger方程在Sobolev空间Hs(R)中,当s < 21时是不适定的,这一结论被人们所熟知.同时,著名团队Colliander,Keel,Sta?lani,Takaoka 和Tao (I-term) 在2002年获得了该方程当指标s > 21时的整体适定性.关于临界空间H21(R)的整体适定性问题一直公开.本章第二节,我们应用带部分修正技术的I方法对这个问题给出了肯定地回答.本章第三节,我们进一步运用这一技术研究周期和非周期情形的质量临界的Schr¨odinger方程,并获得了方程在Hs(R)(或Hs(T)),s > 25时的整体适定性.这一结果改进了Bourgain(菲尔兹奖获得者)和I-term的结论。
第四章主要研究散焦情形的三维库仑位势的波动方法的整体适定性.本章中,我们证明:当初值属于Hs(R3)×Hs?1(R3),当s > 0.7时,方程的整体适定性.我们所获得的结果改进了Kenig,Ponce和Vega (2000),Gallagher和Planchon (2003),Bahouri和Chemin (2006)以及Roy(2009)的工作.这一改进主要来自于我们获得了一个更优的非线性光滑估计.
第五章,我们研究了三维散焦形H12次临界的Hartree方程.运用I方法,几乎Morawetz估计等技术,我们获得了方程低于能量空间的整体适定性和散射性质.这一结论改进了Miao,Xu和Zhao(2009)的工作.
第六章,我们研究Benjamin方程.这是一个不具有伸缩不变解的方程.基于I方法,这类方程的整体适定性主要有两个要素决定:第一,修正能量的误差估计;第二,局部生命跨度.在这一章中,我们证明了一种特殊的双线性估计(这一技术的思想来源于第二章Schr¨odinger-Korteweg-de Vries系统整体适定性的研究),从而延长局部生命跨度.最终我们证明了当初值属于Hs(R)中,当s > ?43时的整体适定性.
第七章,我们从一类五阶KdV方程的特点出发,研究这类方程所含非线性项的双线性估计.首先,我们揭示了该非线性项,基于通常Bourgain空间的双线性估计并不理想(指标s < 14时,不成立).为了进一步的研究,我们构建了修正的Bourgain空间,并证明了基于这类修正的Bourgain空间的双线性估计,从而证明了这类五阶KdV方程的局体适定性.最后,我们证明一个不适定性结论,从而揭示我们所获得的局部适定性结论是最优的.
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.
引文
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