两类非线性色散方程解的动力学行为研究

发布时间：2018-05-01 11:41

本文选题：Davey-Stewartson系统 + 非局部项　；参考：《中国工程物理研究院》2016年博士论文

【摘要】：本论文致力于利用Littlewood-Paley理论,集中紧致方法,变分刻画等现代调和分析工具来研究带有非局部非线性项的两类色散方程的动力学行为,主要包括解的适定性,散射和爆破理论。其中散射理论的研究源于Segal[84]中的一个猜想。很久以来,非线性色散方程的散射理论一直是现代偏微分方程研究领域中备受关注并被广泛研究的问题,可参见Cazenave[11]和Tao[87]的专著等。从物理的角度来看,散射理论是科学家们研究和探测微观自然界的一种非常有效的方法,对量子力学、化学、生物学等诸多自然科学的发展都具有重要的促进作用。如导致发现DNA的X射线晶体学,水波、激波的传播和衰减,X线断层摄影术和利用声纳对水下物体进行的探测等等,这些都需要通过研究粒子或波的散射性质来获得。从数学的角度来看,散射理论主要研究的是非线性色散方程Cauchy问题解的长时间行为。具体地说,在非线性色散方程Cauchy问题解整体存在的前提下,研究当时间t趋于无穷大时,非线性色散方程解在某种范数意义下是否可以被相应的自由方程解所逼近。本文共分六章：第一章为引言,我们以Schrodinger方程为例介绍散射理论及介绍本论文所要研究的两类具有非局部非线性项的混合方程的研究背景及其进展。第二章主要介绍了一些预备知识和一些己知的结果。第三章到第六章我们所考虑的都是三维情形。第三章研究一类修正Davey-Stewartson方程不同参数下解的动力学行为。第四章研究一类修正的非聚焦具有能量临界项的Davey-Stewartson方程解的散射理论。第五章研究一类广义的非聚焦Davey-Stewartson方程解的散射理论。第六章研究了具有卷积型非线性项的混合Schrodinger方程在能量空间中径向解的散射与爆破理论。具体内容如下：第一章为引言,我们以Schrodinger方程为例介绍散射理论及介绍本论文所要研究的两类具有非局部非线性项的混合方程的研究背景及其进展。第二章是预备知识,规定了本文用到的一些记号与定义,并介绍了一些调和分析中的基本理论。第三章主要系统地研究了如下修正Davey-Stewartson系统的Cauchy问题解的适定性,散射与爆破理论,其中在不同参数(λ1,A2)条件下,我们在能量空间中对于方程解的局部和整体适定性,爆破和散射理论给出一个完整的刻画。所用到的方法主要是T. Tao, M. Visan和X. Zhang文献中的扰动理论和Glassey在文献[36]给出的凸性分析。第四章主要利用Kenig和Merle文献[43]的集中紧方法来研究如下修正Davey-Stewartson系统的Cauchy问题解的整体适定性和散射理论,其中主要的困难是方程不保持尺度变换不变性,相互作用的Morawetz估计的失败和非局部项E1(|u|2)u的不对称性。第五章仍然利用Kenig-Merle文献[43]的集中紧方法来研究如下广义的三维Davey-Stewartson系统的Cauchy问题解的整体适定性和散射理论,其中主要的困难是相互作用的Morawetz估计的失败和非局部项E1(|u|2)u的不对称性。第六章我们研究了具有卷积项的混合Schrodinger方程的Cauchy问题的解在能量空间H1(R3)的散射和爆破理论。我们首先利用变分法给出爆破与散射的门槛。然后利用集中紧方法得到散射理论,利用凸性方法得到爆破结果。我们主要克服了来自方程不保持尺度变换不变性和卷积型的非局部项所带来的困难。我们的结果表明在能量空间中聚焦的能量临界项-|u|4u让在解的散射门槛中起着决定性的作用。
[Abstract]:This thesis is devoted to using the Littlewood-Paley theory, the concentrated compact method, the variational portrayal and other modern harmonic analysis tools to study the dynamic behavior of the two classes of dispersion equations with nonlocal nonlinear terms, including the conjectures, scattering and blasting theory of solutions. The scattering theory is derived from a conjecture in Segal[84]. Since the scattering theory of the nonlinear dispersion equation has been a concern and widely studied in the field of modern partial differential equations, we can see the monographs of Cazenave[11] and Tao[87]. From the physical point of view, the scattering theory is a very effective method for scientists to study and detect microcosmic self boundary, and to the quantum force. The development of many natural sciences, such as science, chemistry and biology, has an important role to promote. Such as the discovery of the X ray crystallography of DNA, the propagation and attenuation of water waves, shock waves, X-ray tomography, and sonar detection of underwater objects, etc., all of which need to be obtained by studying the scattering properties of particles or waves. From the point of view, the scattering theory mainly deals with the long time behavior of the solution of the nonlinear dispersion equation Cauchy problem. Specifically, when the solution of the nonlinear dispersion equation Cauchy problem is integral, when the time t tends to infinity, the solution of the nonlinear dispersion equation can be solved by the corresponding free equation solution in a certain number sense. This paper is divided into six chapters: the first chapter is an introduction. We take the Schrodinger equation as an example to introduce the scattering theory and introduce the research background and progress of the two kinds of mixed equations with nonlocal nonlinear terms in this paper. The second chapter mainly introduces some preparatory knowledge and some known results. The third to sixth chapters. All we consider is the three-dimensional case. The third chapter studies the dynamic behavior of a class of solutions with different parameters of the modified Davey-Stewartson equation. The fourth chapter studies the scattering theory of a modified Davey-Stewartson equation with a non focused energy critical term. The fifth chapter studies the dispersion of the solution of a class of generalized non focused Davey-Stewartson equations. The sixth chapter studies the scattering and blasting theory of the radial solution of the mixed Schrodinger equation with a convolution type nonlinear term in the energy space. The specific contents are as follows: the first chapter is an introduction. We take the Schrodinger equation as an example to introduce the scattering theory and introduce the two kinds of nonlocal nonlinear terms in this paper. The research background and progress of the equation. The second chapter is the preparatory knowledge, defines some marks and definitions used in this paper, and introduces some basic theories in the harmonic analysis. The third chapter mainly studies the proper qualitative, scattering and blasting theory of the solution of the Cauchy problem of the Davey-Stewartson system as follows. Under the condition of A2), in the energy space, we give a complete characterization of the local and global fitness of the solutions of the equations, the theory of blasting and scattering. The methods used are the perturbation theory in the literature of T. Tao, M. Visan and X. Zhang and the convexity given by Glassey in the literature [36]. The fourth chapter mainly uses Kenig and Merle document [43]. The centralization method is used to study the overall fitness and scattering theory of the Cauchy solution to the Davey-Stewartson system as follows. The main difficulty is that the equation does not maintain the scale transformation invariance, the Morawetz estimation of the interaction is failed and the non local term E1 (|u|2) u is unsymmetrical. The fifth chapter still uses the Kenig-Merle document [43]. The centralization method is used to study the global fitness and scattering theory of the solution of the Cauchy problem in a generalized three-dimensional Davey-Stewartson system. The main difficulties are the failure of the Morawetz estimation of the interaction and the asymmetry of the non local term E1 (|u|2) U. In the sixth chapter, we study the Cauchy question of the mixed Schrodinger equation with convolution terms. The solution of the problem in the energy space H1 (R3) scattering and blasting theory. Firstly, we use the variational method to give the threshold of blasting and scattering. Then we use the centralized method to get the scattering theory and use the convexity method to get the blasting results. We mainly overcome the non local term which the equation does not maintain the scale change invariance and the convolution type. It is difficult. Our results show that the energy critical term -|u|4u in the energy space plays a decisive role in the scattering threshold of the solution.

【学位授予单位】：中国工程物理研究院
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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