外力场中的Hartree方程的动力学研究

Dynamics of the Hartree Equations in the External Force Field

作者: 专业:数学 导师:邹文明 年度:2013 学位:博士 

关键词
Hartree方程 适定性 基态解 轨道稳定性 相干态

Keywords
Hartree equation, well-posedness, ground states, orbital stability, coherentstates
        Hartree方程由Hartree于1928年提出,是Hartree-Fock理论的重要组成部分,在众多领域有着广泛的应用,自上世纪70年代以来一直得到数学家们的广泛关注。至今,对经典的Hartree方程理论研究已取得丰硕的成果。本文旨在把自由Hartree方程的研究结果推广到有外力作用的Hartree方程中,研究内容涉及到外力场中的Hartree方程的整体适定性、散射理论、驻波解的稳定性及相干态等,这也是我在博士期间发表的文章[4]和[5]的总结和提炼。全文分为三大部分:第一部分研究带有外势的Hartree方程的适定性和散射理论,主要内容如下:1.讨论带有外势的Hartree方程在Σ空间中的的整体适定性,利用Strichartz估计得到局部解的存在性,建立爆破准则,然后利用质量守恒律、能量满足的恒等式、Virial恒等式建立方程在Σ空间中的整体适定性;2.对于上述得到的唯一的Σ解,利用数学归纳法和靴带引理建立其高阶加权Sobolev范数的指数增长性;3.通过建立两个重要恒等式讨论能量次临界时散焦的Hartree方程的散射性质。第二部分研究带有外势的Hartree方程的驻波解及相应的椭圆方程的基态解。具体而言,1.讨论散焦的Hartree方程驻波解的存在性问题;2.通过求解三类变分问题得到聚焦的定态Hartree方程的基态解,并将讨论基态解的正则性,衰减性,对称性和正性等;3.通过上面得到的三个基态解建立含时的Hartree方程驻波解的轨道稳定性。最后一部分构造当初值为半经典波包(相干态)时,半经典Hartree方程的近似解,研究解在传播过程中的行为,包括:1.当卷积的核为有界函数时,构造方程在L2空间中的近似解,从而推广[1]中的结论;2.利用解的高阶Sobolev指数增长性和L2模估计,证明对次临界和临界态的半经典Hartree方程,波包的传播性质与线性情形类似。同时也将讨论初值为两个波包叠加时方程的非线性叠加原理。
    Hartree equation initially proposed by Hartree. As an important part of Hartree-Fock theory, it has a wide range of applications in diferent fields. Since1970s, it hasbeen widespread by mathematicians, fruitful results in theory have been achieved on theclassical Hartree equations.In this thesis, we try to extend the results on the free Hartree equations to the casein the extern fields. It is concerned with well-posedness, scattering, stability of standingwaves and coherent states etc. It is also the summarization and extraction of my papers[4] and [5] published during my ph. D. research.This thesis is divided into three parts: in the first part, we study the well-posednessand scattering for the Cauchy problem of the Hartree equations with potential. For details,1. we discuss the well-posedness of the Hartree equations with potential. The localexistence of the solution shall be established using Strichartz estimates. Then weshow the local Σ solution is global by means of blow-up alternative, mass conser-vation law, the identity the energy satisfied and Virial identity;2. applying the induction and bootstrap argument, we establish the exponential controlof the growth of higher order Sobolev norms for the global Σ solution;3. by establishing two identities, we prove the scattering of the defocusing Hartreeequation in energy subcritical case.In the second part, the standing waves of the Hartree equations with confining po-tential and the ground states of steady focusing equations are considered. We shall1. discuss the existence of the standing waves of defocusing Hartree equation;2. show the existence of ground states of the elliptic equations in the focusing case,meanwhile the regularity, decay, symmetry and positivity of the ground states shallbe discussed.3. establish the orbital stability of the standing waves by means of the stability of theabove ground states.In the last part, we shall construct the approximation solutions of the Hartree equations insemi-classical limit when the initial data is semi-classical wave packets (coherent states)and study the propagation of wave packets. For details,1. we get the approximation solutions in L2of the Hartree equations with smooth kernel. The result in [1] can be extended to general cases;2. the subcritical and critical cases for the Hartree quations with homogeneous k-ernel is studied. By the growth of the solutions and L2estimates, we show thepropagation of wave packets is the same as the linear case. Meanwhile, nonlinearsuperposition principle for two nonlinear wave packets is considered.
        

外力场中的Hartree方程的动力学研究

摘要3-4
Abstract4-5
主要符号对照表8-9
第1章 引言9-21
    1.1 选题背景9-10
    1.2 研究现状10-18
        1.2.1 Hartree方程的整体适定性、散射与爆破10-15
        1.2.2 Hartree方程的驻波解及基态15-17
        1.2.3 半经典Hartree方程及相干态17-18
    1.3 本文的主要工作18-20
    1.4 本文的结构20-21
第2章 预备知识21-27
    2.1 基本不等式21-22
        2.1.1 Young不等式21
        2.1.2 Gagliardo-Nirenberg不等式21
        2.1.3 Hardy-Littlewood-Sobolev不等式21
        2.1.4 重排不等式21-22
    2.2 Strichartz估计22-25
        2.2.1 经典的Strichartz估计23
        2.2.2 Strichartz估计的推广23-25
    2.3 基本引理25-27
        2.3.1 靴带引理25
        2.3.2 Gronwall引理25
        2.3.3 紧嵌入引理25-26
        2.3.4 Profile分解26-27
第3章 带有外势的Hartree方程的整体适定性与散射理论27-38
    3.1 Hartree方程柯西问题的整体适定性27-32
        3.1.1 L~2次临界时解的整体存在唯一性27-29
        3.1.2 能量次临界时方程的整体适定性29-31
        3.1.3 定理3.1的证明31-32
    3.2 解的高阶Sobolev范数的增长性32-34
        3.2.1 预备工作32-33
        3.2.2 解的Σ~k范数的指数增长性33-34
    3.3 散射理论34-38
        3.3.1 预备工作35-36
        3.3.2 能量次临界时散焦方程的散射性质36-38
第4章 带有外势的Hartree方程驻波解的轨道稳定性38-62
    4.1 预备工作38-40
    4.2 椭圆方程的正则性40-42
    4.3 基态解及驻波解的轨道稳定性42-45
        4.3.1 基态解的存在性及其性质42-44
        4.3.2 L~2次临界的聚焦方程驻波解的轨道稳定性44-45
    4.4 ω基态解及驻波解的轨道稳定性45-59
        4.4.1 ω基态解的定义45
        4.4.2 ω >0时基态解的存在性及其性质45-47
        4.4.3 ω→∞时能量次临界的聚焦方程驻波解的轨道稳定性47-55
        4.4.4 ω >-λ0时基态解的存在性及其性质55-57
        4.4.5 ω→-λ0时能量次临界的聚焦方程驻波解的轨道稳定性57-59
    4.5 能量次临界时散焦方程驻波解的存在性59-62
第5章 半经典Hartree方程的相干态及Ehrenfest时间62-86
    5.1 预备工作62-65
        5.1.1 哈密顿系统与拉格朗日作用量62-63
        5.1.2 线性薛定谔方程的L2近似解63-64
        5.1.3 次临界、临界与超临界的分类64-65
    5.2 K ∈ W~(3,∞)(R~d;R)且在原点附近光滑的情形65-72
        5.2.1 L~2解的整体存在性66-67
        5.2.2 近似解的构造67-70
        5.2.3 定理5.1的证明70-72
    5.3 K = |x|~(-γ),0< γ 72-86
        5.3.1 有限时间的 估计73-76
        5.3.2 解在次临界情形时的长时间行为76-78
        5.3.3 定理5.2的证明78-80
        5.3.4 非线性叠加性质80-86
第6章 总结86-88
    6.1 本文的创新点86-87
    6.2 未来工作的展望87-88
参考文献88-95
致谢95-97
个人简历、在学期间发表的学术论文与研究成果97


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