In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t −3 local uniform decay rate (Price's law [54]) for linear waves. As a corollary, we also prove Price's law for certain small perturbations of the Kerr metric.This result was previously established by the second author in [64] on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.
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.
We examine solutions to semilinear wave equations on black hole backgrounds
and give a proof of an analog of the Strauss conjecture on the Schwarzschild
and Kerr, with small angular momentum, black hole backgrounds. The key
estimates are a class of weighted Strichartz estimates, which are used near
infinity where the metrics can be viewed as small perturbations of the
Minkowski metric, and a localized energy estimate on the black hole background,
which handles the behavior in the remaining compact set.Comment: 21 pages, no changes in contents, fix a technical problem in pdf file
it generate
Abstract. We study the dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of this paper is to establish Strichartz estimates for solutions of the aforementioned equation. This follows a local energy decay result for the Kerr space-time obtained in earlier work of Tataru and the author, and uses the techniques and results by the author and collaborators (2010). As an application, we then prove global well-posedness and uniqueness for the energy critical semilinear wave equation with small initial data.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.