Strunk, Nils: Critical well-posedness results for nonlinear Schrödinger equations on compact manifolds. 2015
Inhalt
- Contents
- Introduction
- 1 Basics
- 1.1 Notation
- 1.2 Function spaces and the Fourier transform
- 1.2.1 Lp-spaces and Sobolev spaces
- 1.2.2 The Schwartz class and the Fourier transform
- 1.2.3 The spaces Up and Vp
- 1.3 Fourier series and exponential sums
- 1.4 Riemannian manifolds
- 1.5 Dispersion
- 2 Local and small data global well-posedness
- 2.1 Preliminary remarks
- 2.2 A conditional local and small data global well-posedness result
- 2.3 Rectangular tori in three dimensions
- 2.3.1 Selected results
- 2.3.2 Set-up
- 2.3.3 Linear Strichartz estimates
- 2.3.4 Almost orthogonality
- 2.3.5 The trilinear Strichartz estimate
- 2.4 Rectangular tori in two dimensions
- 2.5 Product of spheres
- 2.5.1 Selected results
- 2.5.2 Set-up
- 2.5.3 A trilinear estimate for spherical harmonics
- 2.5.4 Two exponential sum estimates
- 2.5.5 Almost orthogonality
- 2.5.6 The trilinear Strichartz estimate
- 2.6 Further results on other manifolds and remarks
- 3 Global well-posedness for large data
- 3.1 Set-up and main result
- 3.2 Basic definitions and statements
- 3.3 Local well-posedness and stability theory
- 3.3.1 Estimates on the Duhamel term
- 3.3.2 Local well-posedness
- 3.3.3 Small data global well-posedness
- 3.3.4 Stability
- 3.4 Euclidean profiles
- 3.4.1 Global well-posedness on the Euclidean space
- 3.4.2 Connection between solutions on tori and Euclidean solutions
- 3.5 Profile decomposition
- 3.6 Proof of the main theorem
- 3.7 Further remarks
- Summary
- Bibliography