Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb {T}^{2}$

Sebastian Herr ; Beomjong Kwak

Forum of Mathematics, Pi, Tome 12 (2024)

Résumé

The optimal $L^4$-Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus $\mathbb {T}^2$ is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger $L^4$ bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in $H^s(\mathbb {T}^2)$ for any $s>0$ and data that are small in the critical norm.

Détail

DOI : 10.1017/fmp.2024.11

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