Strichartz estimates for the wave equation on a 2d model convex domain

TL;DR

This paper improves Strichartz estimates for the wave equation on a 2D convex domain by leveraging space-time localization of caustics, leading to better bounds than previously known.

Contribution

It introduces enhanced parametrix constructions and demonstrates improved Strichartz estimates for the wave equation on convex domains, surpassing earlier dispersion-based results.

Findings

Improved Strichartz estimates for 2D convex domains

Enhanced parametrix construction with better localization properties

Independent interest in new analytical techniques for wave equations

Abstract

We prove better Strichartz type estimates than expected from the (optimal) dispersion we obtained in our earlier work on a 2d convex model. This follows from taking full advantage of the space-time localization of caustics in the parametrix we obtain, despite their number increasing like the inverse square root of the distance from the source to the boundary. As a consequence, we improve known Strichartz estimates for the wave equation. Several improvements on our previous parametrix construction are obtained along the way and are of independent interest for further applications.