Strichartz Estimates on Schwarzschild Black Hole Backgrounds

Abstract

Abstract:We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of [29], to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.

Strichartz estimates on Kerr black hole backgrounds

“…Theorem 1.2 follows from Theorem 1.1 exactly as in Section 5 of [13]; hence we will now focus on proving the latter.…”

“…The key to the proof is the approach developed in [14] for small perturbations of the Minkowski metric and in [13] for the Schwarzschild metric which allows one to establish global-in-time Strichartz estimates provided that a strong form of the local energy estimates holds.…”

“…We first improve this to a stronger result that only requires logarithmic losses in the energy norm. We then apply the techniques from [14] and [13] to obtain Strichartz estimates for all nonsharp exponents.…”

“…The role of ψ is to make sure that b ps = 1 when τ λ a and τ λ a . Observe that, as opposed to their Schwarzschild counterparts a ps and a −1 ps defined in [13], the symbols b ps and b −1 ps will depend on τ and Φ.…”

“…We start with the following estimate near r = 3M , which is the equivalent of Proposition 3.3 from [13]:…”