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Well-posedness and Scattering Theory for Fourth-order Schr(?)dinger Equations

Author: ZhengJiQiang
Tutor: MiaoChangXing
School: Chinese Academy of Engineering Physics
Course: Basic mathematics
Keywords: Fourth order Schr (o ¨) dinger equation Posedness [k, Z] - multiplier mold Strichartz estimates Scattering theory
CLC: O175
Type: Master's thesis
Year: 2011
Downloads: 10
Quote: 0
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Abstract


This paper mainly study consists of two parts : the first part Bourgain [ 6 ] , Tao [ 12 ] perturbation method with combining items , fourth-order scattering theory of Schrodinger equation ; second part of the use of Tao [ 39 ] of [ K ; Z posedness create a class of fourth-order Schrodinger equation ] - multiplier mode method . example to introduce the basic content of the scattering theory Schrodinger equation in the first chapter , the first section II introduced Bourgain space definition and it some basic properties , and then introduce the Bourgain space linear estimation in the second chapter , we use Bourgain [ 6 ] , Tao [ 12 ] in the perturbation method to study with combining items , fourth-order scattering theory of Schrodinger equation . fourth-order Schrodinger Fangcheng Yuan in the Karpman-Shagalov [19,20] intense laser beams . prove the main idea is as follows : the first equation by perturbation theory has good local posedness resulting local solution the existence interval only depends on the initial value of the Hx2 mode rather than depend on the behavior of the initial value . then the use of the good local posedness combined to control the global existence of the solution can be launched on the overall kinetic energy . nonlinear terms are non- focused interaction Morawetz estimate [ 30,36 ] can be introduced to the whole space-time mode boundedness resulting scattering ; subcritical case focused sufficiently small quality using quality sufficiently ensure that sub-critical items in a certain norm is a small amount , and then the whole space-time solution by perturbation theory to estimate the ultimate scattering theory . third chapter , we first standard fixed point theory , local posedness problem boils down is the bilinear estimated ; then introduced Tao [ 39 ] of [ K ; Z - multiplier mode theory basic framework , and gives [ K ; Finally, [ Z ] - some of the basic properties of the multiplier mode ; K ; Z - multiplier theory to prove bilinear estimates .

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations
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