Global Well-Posedness of the Cubic Nonlinear Schrödinger Equation on Closed Manifolds

Abstract

We consider the defocusing cubic non-linear Schrödinger equation on general closed (compact without boundary) Riemannian surfaces. The problem was shown to be locally well-posed in H s (M) for in [8 Burq , N. , Gerard , P. , Tzvetkov , N. ( 2004 ). Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds . Amer. J. Math. 126 : 569 – 605 .[Crossref], [Web of Science ®] , [Google Scholar]]. 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 . This generalizes, without any loss in regularity, the results in [6 Bourgain , J. ( 2004 ). A remark on normal forms and the I-method for periodic NLS . J. Anal. Math. 94 : 125 – 157 .[Crossref], [Web of Science ®] , [Google Scholar], 18 De Silva , D. , Pavlovic , N. , Staffilani , G. , Tzirakis , N. ( 2007 ). Global well-posedness for a periodic nonlinear Schrdinger equation in 1D and 2D . Discrete Contin. Dyn. Syst. 19 : 37 – 65 .[Crossref], [Web of Science ®] , [Google Scholar]], where the same result is proved for the torus 𝕋2. The proof relies on the I-method of Colliander et al. [17 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2002 ). Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation . Math. Res. Letters 9 : 659 – 682 .[Crossref], [Web of Science ®] , [Google Scholar]] a semi-classical bilinear Strichartz estimate proved by the author in [22 Hani , Z. ( 2011 ). A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds. To appear in Analysis and PDE. [Google Scholar]], and spectral localization estimates for products of eigenfunctions, which is essential to develop multilinear spectral analysis on general compact manifolds.