Defocusing Nonlinear Schrodinger Equations

✍ Scribed by Benjamin Dodson

• Publisher: Cambridge University Press
• Year: 2019
• Tongue: English
• Leaves: 255
• Series: Cambridge Tracts in Mathematics 217
• Category: Library

✦ Synopsis

This study of Schrödinger equations with power-type nonlinearity provides a great deal of insight into other dispersive partial differential equations and geometric partial differential equations. It presents important proofs, using tools from harmonic analysis, microlocal analysis, functional analysis, and topology. This includes a new proof of Keel–Tao endpoint Strichartz estimates, and a new proof of Bourgain's result for radial, energy-critical NLS. It also provides a detailed presentation of scattering results for energy-critical and mass-critical equations. This book is suitable as the basis for a one-semester course, and serves as a useful introduction to nonlinear Schrödinger equations for those with a background in harmonic analysis, functional analysis, and partial differential equations.

✦ Table of Contents

Contents......Page 8
Preface......Page 10
Acknowledgments......Page 13
1.1 Linear Schrodinger Equation and Preliminaries......Page 14
1.2 Strichartz Estimates......Page 23
1.3 Small Data Mass-Critical Problem......Page 37
1.4 A Large Data Global Well-Posedness Result......Page 46
2.1 The Cubic NLS in Three and Four Dimensions with Small Data......Page 54
2.2 Scattering for the Radial Cubic NLS in Three Dimen- sions with Large Data......Page 60
2.3 The Radially Symmetric, Cubic Problem in Four Dimensions......Page 65
3.1 Small Data Energy-Critical Problem......Page 77
3.2 Profile Decomposition for the Energy-Critical Problem......Page 85
3.3 Global Well-Posedness and Scattering When d ≥5......Page 98
3.4 Interaction Morawetz Estimate......Page 109
4.1 Bilinear Estimates......Page 115
4.2 Mass-Critical Profile Decomposition......Page 122
4.3 Radial Mass-Critical Problem in Dimensions d ≥2......Page 141
4.4 Nonradial Mass-Critical Problem in Dimensions d ≥3......Page 159
5.1 The Energy-Critical Problem in Dimensions Three and Four......Page 177
5.2 Three-Dimensional Energy-Critical Problem......Page 191
5.3 The Mass-Critical Problem When d = 1......Page 204
5.4 The Two-Dimensional Mass-Critical Problem......Page 227
References......Page 249
Index......Page 254