Bourgain, J. On Strichartz’s inequalities and the nonlinear Schrödinger equation on irrational tori. (English) Zbl 1169.35054 Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. Lectures of the CMI/IAS workshop on mathematical aspects of nonlinear PDEs, Princeton, NJ, USA, 2004. Princeton, NJ: Princeton University Press (ISBN 978-0-691-12955-6/pbk; 978-0-691-12860-3/hbk). Annals of Mathematics Studies 163, 1-20 (2007). The author establishes Strichartz estimates for the linear counterpart of the nonlinear Schrödinger equation on irrational tori in three space dimensions. To this aim he combines the Hardy-Littlewood circle method with the Fourier-analytical approach from the Euclidean case. The obtained estimates are then applied to derive local and global wellposedness results for the Cauchy problem for the defocusing nonlinear Schrodinger equation on a three dimensional irrational torus (the cubic case is included). The use of the author’s \(X_{s,b}\) spaces is also needed to obtain the required estimates for the nonlinear terms.For the entire collection see [Zbl 1113.35005]. Reviewer: Catalin Popa (Iaşi) Cited in 22 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35J10 Schrödinger operator, Schrödinger equation 58J05 Elliptic equations on manifolds, general theory 35B45 A priori estimates in context of PDEs Keywords:nonlinear Schrödinger equation; irrational torus; Strichartz estimate; well-posed Cauchy problem PDF BibTeX XML Cite \textit{J. Bourgain}, Ann. Math. Stud. 163, 1--20 (2007; Zbl 1169.35054) OpenURL