Strichartz estimates for the fractional Schr\"odinger and wave equations on compact manifolds without boundary

TL;DR

This paper establishes Strichartz estimates for fractional Schr"odinger and wave equations on both Euclidean spaces and compact manifolds, enabling analysis of nonlinear equations in these geometric settings.

Contribution

It proves new Strichartz estimates on compact Riemannian manifolds for fractional Schr"odinger and wave equations, extending previous Euclidean results.

Findings

Strichartz estimates on $\

Applications to local well-posedness of nonlinear equations.

Abstract

We firstly prove Strichartz estimates for the fractional Schr\"odinger equations on $\mathbb{R}^d$ endowed with a smooth bounded metric $g$. We then prove Strichartz estimates for the fractional Schr\"odinger and wave equations on compact Riemannian manifolds without boundary $(M,g)$. We finally give applications of Strichartz estimates for the local well-posedness of the pure power-type nonlinear fractional Schr\"odinger and wave equations posed on $(M,g)$.