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求是数学短期课程(19-20学年秋冬学期)求是数学短期课程Introduction to the Nonlinear Schroedinger Equation

编辑:wfy 时间:2019年09月20日 访问次数:1172

求是数学短期课程(19-20学年秋冬学期)

求是数学短期课程

Introduction to the Nonlinear Schroedinger Equation

主讲人:法国索邦大学(巴黎六大)Thierry  Cazenave教授

授课时间和地点:

10月9日周三晚上: 18:00-20:15(3课时),玉泉工商楼105

10月11日周五晚上:18:00-20:15(3课时),玉泉工商楼105

10月12日周六下午: 13:00-15:15(3课时),玉泉外经贸楼113

10月14日周一下午: 13:30-15:45(3课时),玉泉外经贸楼113

10月16日周三晚上: 18:45-20:15(3课时),玉泉工商楼105

本课程15个课时。适合于数学学院数学与应用数学专业四年级及部分三年级本科生,以及研究生选修。课程结束后的作业或试卷交助教,报告交杨利平老师,需要学分的同学在四年级第二学期的选课时选择专题讲座(抵8次学术讲座),由教务秘书登记成绩。

课程简介

1Sobolev spaces on R^N(3 lectures)

a Definitions and basic properties (approximation by smooth functions)

b The Fourier transform and Sobolev / Besov spaces

c The chain rule

d Sobolev and Gagliardo-Nirenberg inequalities

e Compactness properties: local Rellich, and profile decomposition (if we have

time)

f Ground states and best constants in certain Gagliardo-Nirenberg's inequalities

2The linear Schroedinger equation (2 lectures)

a The fundamental solution and the Schroedinger group

b Dispersive and Strichartz estimates

c The nonhomogeneous equation and Duhamel's formula,

3Local theory for the nonlinear Schroedinger equation (NLS) (2 lectures)

a Local theory in L^2

b Local theory in H^1

4NLS in the defocusing case (2 lectures)

a Global existence

b Asymptotic completeness and scattering

5NLS in the focusing case (3 lectures)

a Low-energy scattering

b The viriel identity and finite-time blowup

c A Schroedinger equation with nonlinear source term: construction of blow-up solutions