A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes

Abstract

We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner-Nordström black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds. States,

On global dynamics of the Maxwell–Klein–Gordon equations

“…Thus, this idea should be viewed as a vector field method version of conformal transformations. We also remark here that using such weighted vector fields with order 2 has been used in the works [1], [2] of Angelopoulos-Aretakis-Gajic for deriving the sharp decay of linear waves on black hole spacetimes. However in those works, commuting the equation with the vector field r 2 L is straightforward when writing the equation in terms of the radiation field rφ under a suitable null frame.…”

“…We only derive the bound onρ sinceσ can be bounded exactly in the same manner. First of all, for l = (0, 1, 0) and k = (0, 2, 0), we have L L Ω (rρ) = r −1 L(r 2 ρ (1) ) − ρ (1) .…”

Generic Blow-Up Results for the Wave Equation in the Interior of a Schwarzschild Black Hole

“…Remark 1.15. The notion of genericity of smooth solutions in the previous theorem and in the subsequent corollary comes from [4,5,6]. The initial data on Σ are assumed to have finite energies of the form (4.1), for a certain finite number of derivatives and r-weights.…”

“…The authors would like to thank Mihalis Dafermos for valuable discussions. They would also like to thank Yannis Angelopoulos, Stefanos Aretakis, Dejan Gajic for helpful comments on their recent series of papers [4,5,6]. G.F. was supported by the EPSRC grant EP/K00865X/1 on 'Singularities of Geometric Partial Differential Equations'.…”

On the link between the Maxwell and linearised Einstein equations on Schwarzschild

“…4 Remark that the improved results of [AAG18] for decay of the scalar wave equation are needed to even ascend the hierarchy so as to obtain the weak decay estimates for γ in question.…”