Strichartz-type Estimates for Wave Equation for Normally Hyperbolic Trapped Domains

TL;DR

This paper proves Strichartz-type estimates for wave equations on manifolds with normally hyperbolic trapped obstacles, revealing derivative loss but controlling forcing terms with mixed Lebesgue norms.

Contribution

It establishes new mixed-norm Strichartz estimates for wave equations in the presence of normally hyperbolic trapped obstacles, including derivative loss considerations.

Findings

Global Strichartz estimate with derivative loss.

Forcing term bounded by two Lebesgue mixed norms.

Extension to exterior of normally hyperbolic trapped obstacles.

Abstract

We establish a mixed-norm Strichartz type estimate for the wave equation on Riemannian manifolds $(\Omega,g)$, for the case that $\Omega$ is the exterior of a smooth, normally hyperbolic trapped obstacle in $n$ dimensional Euclidean space, and $n$ is a positive odd integer. As for the normally hyperbolic trapped obstacles, we will some loss of derivatives for data in the local energy decay estimate. Hence the global Strichartz estimate has a derivative loss. However, we can show that the forcing term is bounded by the sum of no more than two Lebesgue $(p,q)$ mixed norms.