高階Schrodinger方程的Strichartz估計.pdf

1、華中科技大學碩士學位論文高階Schrodinger方程的Strichartz估計姓名：李炫申請學位級別：碩士專業(yè)：應用數(shù)學指導教師：鄭權20090515AbstractSince the 1920s, theory of Schr¨ odinger operators has constantly been a central studied sub-ject of modern mathematics and physics

2、. Moreover, Lp ? Lq estimates, Strichartz time-spaceestimates, local smoothing estimates, weighted estimates, weak global smoothing estimates andmaximal operator’s estimates of Schr¨ odinger equations have been an i

3、mportant topic in the recenttwenty years, because the investigation of this topic has abundant applied backgrounds as wellas very strong theoretic meaning. The study of higher-order Schr¨ odinger operator P(D) + Vwh

4、ich is the generalization of Schr¨ odinger operator ?? + V not only can enrich its contents butalso deepen its understanding. The main purpose of this thesis is to study Strichartz estimatesof higher-order Schr¨

5、; odinger equations with a time-depended potential in the case that P is a realhomogeneous elliptic polynomial. Compared with past relating works, the most important featureof this paper is to deal with the case that P i

6、s a higher-order operator.This thesis consists of two parts: in the first part, backgrounds and recent results of Strichartzestimates of Schr¨ odinger equation are presented, and so are the other related results; in

7、 thesecond part, it proves that the strichartz estimates of the higher-order Schr¨ odinger equation witha time-depended potential V (t, x) ∈ Lr tLs x hold in the case that it gives. Firstly it gives thebackgrounds a

8、nd recently results of the problem, and then it gives my result of the problem. Theresult contains two parts: the first one is about r ∈ [1, ∞), and the problem is studied by usingH¨ older estimates and the contract

9、ion mapping principle; the second one is about r = ∞, and theproblem is studied by using the method of fixing time and community of the potential to the time.Key words:Schr¨ odinger equation， time-depended potential