ALMOST-CONSERVATION-LAWSAND-GLOBAL-ROUGH-SOLUTIONSTO-A-NONLINEAR-SCHR¨ODINGER-EQUATION.pdfVIP

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS ¨ TO A NONLINEAR SCHRODINGER EQUATION J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO Abstract. We prove an “almost conservation law” to obtain global-in-time well-posedness for the cubic, defo- s n 4 , 5 , respectively. cussing nonlinear Schr¨odinger equation in H (R ) when n = 2, 3 and s 7 6 1. Introduction and Statement of Results We study the following initial value problem for a defocussing nonlinear Schr¨odinger equation, 2 n (1.1) i∂ φ(x, t) + ∆φ(x, t) = |φ(x, t)| φ(x, t) x ∈ R , t ≥ 0 t s n (1.2) φ(x, 0) = φ (x) ∈ H (R ) 0 s n when n = 2, 3. Here H (R ) denotes the usual inhomogeneous Sobolev space. Our goal is to loosen the regularity requirements on the initial data which ensure global-in-time solutions. In particular, we aim to extend the global theory to certain infinite energy initial data. 1 1 It is known [5] that (1.1)-(1.2) is well-posed locally in time when n = 2, 3 and s 0, 2 respectively . In addition, these local solutions enjoy L2 conservation