Strichartz estimates for the wave equation on flat cones

TL;DR

This paper establishes dispersive and Strichartz estimates for the wave equation on flat cones, enabling analysis of wave behavior and nonlinear wave equations on conic manifolds and related geometries.

Contribution

It provides explicit dispersive and Strichartz estimates for wave equations on flat cones, extending results to polygons and conic surfaces, and applies these to nonlinear wave well-posedness.

Findings

Dispersive estimates for wave propagation on flat cones

Scale-invariant Strichartz estimates derived

Well-posedness results for nonlinear wave equations on conic manifolds

Abstract

We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius $\rho > 0$, the manifold $\mathbb{R}_+ \times \mathbb{R} / 2 \pi \rho \mathbb{Z}$ equipped with the metric $\g(r,\theta) = dr^2 + r^2 d\theta^2$. Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with conic singularities. This in turn yields well-posedness results for the nonlinear wave equation on such manifolds. Morawetz estimates on the cone are also treated.