Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications

2010/11/02 by Sebastian Herr, Daniel Tataru, Nikolay Tzvetkov · 2 citations
Mathematics · #math.AP #msc:35Q55

paper · pdf · doi:10.1515/crelle-2012-0013

published as J. Reine Angew. Math. (2012), Vol. 2014, No. 690, 65-78 · 15 pages

arxiv created 2010/11/02 · arxiv updated 2014/05/05

Abstract

We consider the energy critical nonlinear Schrödinger equation on periodic domains of the form R^m x T^4-m with m=0,1,2,3. Assuming that a certain L⁴ Strichartz estimate holds for solutions to the corresponding linear Schrödinger equation, we prove that the nonlinear problem is locally well-posed in the energy space. Then we verify that the L⁴ estimate holds if m=2,3, leaving open the cases m=0,1.

Cited by

• Almost sure local well-posedness for the nonlinear Schrödinger equations on \Bbb T^d with non-algebraic nonlinearity
• Hyperbolic nonlinear Schrödinger equations on ℝ× \mathbbT