On refined local smoothing estimates for the Schr\"odinger equation in exterior domains

TL;DR

This paper investigates refined local smoothing estimates for the Schrödinger equation outside convex obstacles, demonstrating how these estimates can lead to Strichartz bounds, especially under Neumann boundary conditions.

Contribution

It establishes that refined local smoothing estimates, valid in exterior convex domains, can be combined with wave packet methods to derive Strichartz estimates for Schrödinger solutions.

Findings

Refined local smoothing estimates can be achieved in exterior convex domains.

These estimates enable the derivation of Strichartz estimates via wave packet constructions.

Application to Neumann boundary conditions extends the scope of known results.

Abstract

We consider refinements of the local smoothing estimates for the Schr\"odinger equation in domains which are exterior to a strictly convex obstacle in $\RR^n$. By restricting the solution to small, frequency dependent collars of the boundary, it is expected that taking its square integral in space-time should exhibit a larger gain in regularity when compared to the usual gain of half a derivative. By a result of Ivanovici, these refined local smoothing estimates are satisfied by solutions in the exterior of a ball. We show that when such estimates are valid, they can be combined with wave packet parametrix constructions to yield Strichartz estimates. This provides an avenue for obtaining these bounds when Neumann boundary conditions are imposed.