On Strichartz’s inequalities and the nonlinear Schrödinger equation on irrational tori. (English) Zbl 1169.35054

Bourgain, J.

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)

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