Abstract
We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.
Highlights
The non-linear Einstein vacuum equations with cosmological constant, Ric [g] = g, (1)constitute a complicated coupled quasi-linear hyperbolic system of partial differential equations for a Lorentzian metric g
(1) for = 0 is a subject of intense investigation [5,6], with the current state-of-theart being the results in [7] establishing the linear stability of Schwarzschild, and [8] establishing nonlinear stability for Schwarzschild within a restricted symmetry class
For Problem 2 it is well-known that exponential decay of energy holds without loss of derivatives, see [29,30,31]. (It is an entertaining exercise to prove this in a more robust fashion using the methods of this paper.) For Problem 1, there will be a derivative loss present in any decay estimate, as seen in Theorem 1.1.7
Summary
We construct a solution of the conformal wave equation in AdS which contradicts estimate (3) of Theorem 1.1 without the ε[∂t ]-term on the right hand side. (1) Problem 1: The wave equation (12) on Rt × S3h (with the natural product metric of wthiethEibnosutenidnacryyliantdψer)=whπ2e,rewSh3heries the (say northern) hemisphere of the 3-sphere S3 (say optimally) dissipative boundary conditions are imposed. (It is an entertaining exercise to prove this in a more robust fashion using the methods of this paper.) For Problem 1, there will be a derivative loss present in any decay estimate, as seen in Theorem 1.1.7 This phenomenon can be explained in the geometric optics approximation for the wave equation.
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