兩類非線性色散方程解的動(dòng)力學(xué)行為研究
本文選題:Davey-Stewartson系統(tǒng) + 非局部項(xiàng); 參考:《中國工程物理研究院》2016年博士論文
【摘要】:本論文致力于利用Littlewood-Paley理論,集中緊致方法,變分刻畫等現(xiàn)代調(diào)和分析工具來研究帶有非局部非線性項(xiàng)的兩類色散方程的動(dòng)力學(xué)行為,主要包括解的適定性,散射和爆破理論。其中散射理論的研究源于Segal[84]中的一個(gè)猜想。很久以來,非線性色散方程的散射理論一直是現(xiàn)代偏微分方程研究領(lǐng)域中備受關(guān)注并被廣泛研究的問題,可參見Cazenave[11]和Tao[87]的專著等。從物理的角度來看,散射理論是科學(xué)家們研究和探測微觀自然界的一種非常有效的方法,對量子力學(xué)、化學(xué)、生物學(xué)等諸多自然科學(xué)的發(fā)展都具有重要的促進(jìn)作用。如導(dǎo)致發(fā)現(xiàn)DNA的X射線晶體學(xué),水波、激波的傳播和衰減,X線斷層攝影術(shù)和利用聲納對水下物體進(jìn)行的探測等等,這些都需要通過研究粒子或波的散射性質(zhì)來獲得。從數(shù)學(xué)的角度來看,散射理論主要研究的是非線性色散方程Cauchy問題解的長時(shí)間行為。具體地說,在非線性色散方程Cauchy問題解整體存在的前提下,研究當(dāng)時(shí)間t趨于無窮大時(shí),非線性色散方程解在某種范數(shù)意義下是否可以被相應(yīng)的自由方程解所逼近。本文共分六章:第一章為引言,我們以Schrodinger方程為例介紹散射理論及介紹本論文所要研究的兩類具有非局部非線性項(xiàng)的混合方程的研究背景及其進(jìn)展。第二章主要介紹了一些預(yù)備知識(shí)和一些己知的結(jié)果。第三章到第六章我們所考慮的都是三維情形。第三章研究一類修正Davey-Stewartson方程不同參數(shù)下解的動(dòng)力學(xué)行為。第四章研究一類修正的非聚焦具有能量臨界項(xiàng)的Davey-Stewartson方程解的散射理論。第五章研究一類廣義的非聚焦Davey-Stewartson方程解的散射理論。第六章研究了具有卷積型非線性項(xiàng)的混合Schrodinger方程在能量空間中徑向解的散射與爆破理論。具體內(nèi)容如下:第一章為引言,我們以Schrodinger方程為例介紹散射理論及介紹本論文所要研究的兩類具有非局部非線性項(xiàng)的混合方程的研究背景及其進(jìn)展。第二章是預(yù)備知識(shí),規(guī)定了本文用到的一些記號與定義,并介紹了一些調(diào)和分析中的基本理論。第三章主要系統(tǒng)地研究了如下修正Davey-Stewartson系統(tǒng)的Cauchy問題解的適定性,散射與爆破理論,其中在不同參數(shù)(λ1,A2)條件下,我們在能量空間中對于方程解的局部和整體適定性,爆破和散射理論給出一個(gè)完整的刻畫。所用到的方法主要是T. Tao, M. Visan和X. Zhang文獻(xiàn)中的擾動(dòng)理論和Glassey在文獻(xiàn)[36]給出的凸性分析。第四章主要利用Kenig和Merle文獻(xiàn)[43]的集中緊方法來研究如下修正Davey-Stewartson系統(tǒng)的Cauchy問題解的整體適定性和散射理論,其中主要的困難是方程不保持尺度變換不變性,相互作用的Morawetz估計(jì)的失敗和非局部項(xiàng)E1(|u|2)u的不對稱性。第五章仍然利用Kenig-Merle文獻(xiàn)[43]的集中緊方法來研究如下廣義的三維Davey-Stewartson系統(tǒng)的Cauchy問題解的整體適定性和散射理論,其中主要的困難是相互作用的Morawetz估計(jì)的失敗和非局部項(xiàng)E1(|u|2)u的不對稱性。第六章我們研究了具有卷積項(xiàng)的混合Schrodinger方程的Cauchy問題的解在能量空間H1(R3)的散射和爆破理論。我們首先利用變分法給出爆破與散射的門檻。然后利用集中緊方法得到散射理論,利用凸性方法得到爆破結(jié)果。我們主要克服了來自方程不保持尺度變換不變性和卷積型的非局部項(xiàng)所帶來的困難。我們的結(jié)果表明在能量空間中聚焦的能量臨界項(xiàng)-|u|4u讓在解的散射門檻中起著決定性的作用。
[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.
【學(xué)位授予單位】:中國工程物理研究院
【學(xué)位級別】:博士
【學(xué)位授予年份】:2016
【分類號】:O175
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