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

2024/07/04 by Qionglei Chen, Yilin Song, Qionglei, Chen +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2407.03908

Abstract

In this article, we investigate the global well-posedness for cubic nonlinear Schrödinger equation(NLS) i∂_tu+Δ_gu=|u|²u posed on the three dimensional compact manifolds (M,g) with initial data u₀∈ H^s(M) where s>(√(21)-1)/(4) for Zoll manifold and s>(1+3√(5))/(8) for the product of spheres \BbbS²×\BbbS¹. 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.

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