Strichartz estimates for partially periodic solutions to Schr\"odinger equations in 4d and applications

TL;DR

This paper investigates Strichartz estimates for the nonlinear Schrödinger equation on partially periodic domains in four dimensions, establishing local well-posedness under certain conditions and verifying these estimates for specific cases.

Contribution

The paper proves local well-posedness of the energy-critical nonlinear Schrödinger equation on R^m x T^{4-m} assuming an L^4 Strichartz estimate, and confirms the estimate for m=2,3.

Findings

L^4 Strichartz estimate holds for m=2,3

Local well-posedness in energy space under the estimate

Open cases for m=0,1

Abstract

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