A new physical-space approach to decay for the wave equation with applications to black hole spacetimes

By Mihalis Dafermos, Igor Rodnianski. 2009

237 citations recorded by the source corpus.

We present a new general method for proving global decay of energy through a suitable spacetime foliation, as well as pointwise decay, starting from an integrated local energy decay estimate. The method is quite robust, requiring only physical space techniques, and circumvents use of multipliers or commutators with weights growing in t. In particular, the method applies to a wide class of perturbations of Minkowski space as well as to Schwarzschild and Kerr black hole exteriors.

Subfields: Astronomy and Astrophysics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/d9ac04c9c1724e2129889f6302bdfd3bc78634b6

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Citations in the corpus

Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009, background): “These include a pointwise decay estimate with rate t−1+ [20], a pointwise in time decay with rate t−3/2+ [36] (see also [18]), a Strichartz estimate [52], decay with rate t−3 [48].”
Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential (2014, methodology): “For the decay estimates in the interior region, we use the pigeonhole argument in [Dafermos and Rodnianski 2010].”
Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars (2014, methodology): “The crucial difference between these stars and a Schwarzschild black hole (which also possesses a photon sphere) is that these geometries do not possess a horizon, and are completely regular down to r = 0.”
Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method (2015, methodology): “While this method provides very precise estimates on the solutions, it does not seem sufficiently robust to be applicable to quasilinear system of wave equations such as the Einstein equations, and the method of choice3 for proving decay in view of such applications is the commutation vector field m”
Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations II (2011, result): “Recent results [ 17 ] overcome these difficulties for the whole subextreme range of Kerr.”
Price's Law on Nonstationary Space-Times (2011, unknown): “We can rewrite the radial three dimensional equation (3.28) as a one dimensional problem (∂2t − ∂2r )(rvα) = rHα which has the solution (rvα)(t, r) = 1 2 ∫ t 0 ∫ r+t−s |r−t+s| ρHα(s, ρ)dρds.”
Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I (2011, methodology): “Two new methods were presented recently for obtaining sharp decay of energy flux and pointwise decay on black hole spacetimes; see [ 26 ,50].”