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

2010/08/17 by Zaher Hani, Hani, Zaher · 1 citation
Mathematics · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1008.2826

openalex publication_date 2010/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥ 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 \T². 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.

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• Long-time instability and unbounded Sobolev orbits for some periodic\n nonlinear Schr "odinger equations

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