Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds

TL;DR

This paper establishes global well-posedness for the cubic nonlinear Schrödinger equation on three-dimensional compact manifolds with rough initial data, extending previous results to non-flat geometries using advanced harmonic analysis techniques.

Contribution

It introduces new multilinear eigenfunction estimates and combines them with the I-method and Strichartz estimates to prove global well-posedness on non-flat compact manifolds.

Findings

Global well-posedness for rough initial data on Zoll manifolds.

Polynomial bounds for the growth of the solution's Sobolev norm.

Extension of previous flat and waveguide results to general compact manifolds.

Abstract

In this article, we investigate the global well-posedness for cubic nonlinear Schr\"{o}dinger equation(NLS) $ i\partial_tu+\Delta_gu=|u|^2u$ posed on the three dimensional compact manifolds $(M,g)$ with initial data $u_0\in H^s(M)$ where $s>\frac{\sqrt{21}-1}{4}$ for Zoll manifold and $s>\frac{1+3\sqrt{5}}{8}$ for the product of spheres $\Bbb{S}^2\times\Bbb{S}^1$. We utilize the multilinear eigenfunction estimate on compact manifold to treat the interaction of different frequencies, which is more complicated compared to the case of flat torus [C. Fan, G. Staffilani, H. Wang, B. Wilson, Anal. PDE, 11 (2018), 919-944.] and waveguide manifold [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.]. Moreover, combining with the I-method adapted to the non-periodic case, bilinear Strichartz estimates along with the scale-invariant $L^p$ linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the $H^s$ norm of solution $u$.