Global Strichartz estimates for the wave equation with a time-periodic non-trapping metric

TL;DR

This paper establishes global Strichartz estimates for wave equations with time-periodic, non-trapping metrics that are perturbations of the Euclidean metric, assuming no resonances with magnitude greater than or equal to one.

Contribution

It provides the first known global Strichartz estimates for wave equations with time-periodic, non-trapping metrics under the resonance-free condition.

Findings

Proves global Strichartz estimates for wave equations with time-periodic metrics.

Demonstrates the importance of non-trapping and resonance-free conditions.

Extends previous results to time-periodic, non-trapping settings.

Abstract

We obtain global Strichartz estimates for the solution $u$ of the wave equation $\partial_t^2 u-\Div_x(a(t,x)\nabla_xu)=0$ with time-periodic metric $a(t,x)$ equal to 1 outside a compact set with respect to $x$. We assume $a(t,x)$ is a non-trapping perturbation and moreover, we suppose that there are no resonances $z_j\in\mathbb{C}$ with $|z_j|\geq1$.