Global well-posedness of the cubic nonlinear Schrödinger equation on compact manifolds without boundary

Abstract

We consider the cubic non-linear Schrödinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\ge 1$ follows easily from conservation of energy and standard arguments. In this work, we extend the range of global well-posedness to $s>2/3$. This generalizes, without any loss in regularity, a similar result on $\T^2$. The proof relies on the I-method of Colliander, Keel, Staffilani, Takaoka, and Tao, a semi-classical bilinear Strichartz estimate proved by the author, and spectral localization estimates for products of eigenfunctions, which is essential to develop multilinear spectral analysis on general compact manifolds.

47 pages. Final version: to appear in CPDE

References in corpus (3)

• Ill-posedness for nonlinear Schrodinger and wave equations
• Global well-posedness of the energy-critical defocusing NLS on $\mathbb{R}\times {\mathbb{T}}^3$
• Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications

Cited by in corpus (4)

• Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations
• Global regularity for the energy-critical NLS on ${\mathbb{S}}^3$
• On scattering for the quintic defocusing nonlinear Schrödinger equation on \R \times \T^2
• Local and global well-posedness for a quadratic Schrödinger system on spheres and Zoll manifolds