薛丁格方程式的局部適定性

On the local-wellposedness of the Schrödinger equation

游步庭(Pu-Ting Yu)

指導教授 ： 江金城

國立清華大學/理學院/數學系/碩士(2014年)

https://doi.org/10.6843/NTHU.2014.00336

摘要

本論文主要是探討 線性的 薛丁格方程式 及波方程式 的 Stric hartz artz artz artz 估計，在介紹完 Strichartz估計，我們針對一組非線性 薛丁格方程式，並應用 Strichartz估計去證明在某些空間下該非線性薛丁格方程式的局部適定性 。

關鍵字

薛丁格

並列摘要

This dissertation is focusing on introducing and proving Strichartz estimates for linear Schrodinger equation and wave equation and their application to the Cauchy problem of Schrodinger equation.
We will use some techniques of real analysis to prove them, so at the rst, we will introduce some tools of real analysis such as some operations, theorems of real analysis, and any tools we will use in order to avoid confusion.
Second, we focus on proving Strichartz estimate for linear Schrodinger equation and wave equation. For Schrodinger eqaution ,the most important tool we apply is Hardy-Littlewood-Sobolev Theorem which helps us transform a property of solutions of Schrodinger equation into a estimate under the so-called mixed norm.
For Wave equation, we focus on R3, and the main idea is we consider homogeneous case with given initial datum and inhomogeneous case but with zero initial datum, respectively, and derive estimates we want respectively, nally combine them and apply Triangle inequality to get the Strichartz Estimate.
And in the last section, we show a application of linear Strichartz estimate for a Schrodinger equation which indicates under some certain conditions and specic spaces, the local well-posedness of a nonlinear Schrodinger equation will be assured.

並列關鍵字

Schrödinger equation

參考文獻

[1] Terence Tao, Nonlinear Dispersive Equations : local and global analysis, CBMS regional conference series in mathematics, July 2006.

[2] F.Linares and G.Ponce, Introduction to Nonlinear Dispersive Equations,Springer, June 2008.

[4] Gerald B.Folland, Fourier analysis and its applications,

[3] Christopher D.Sogge, Lecture on Nonlinear Wave Equations,international Press, February 2008.

Providence, R.I.: American Mathematical Society, January 2009.