Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space

Blue, Pieter; Sterbenz, Jacob

doi:10.1007/s00220-006-0101-6

Cited by 125 publications

(233 citation statements)

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“…The decay argument presented here departs from earlier work that either makes use of multipliers with weights in the temporal variable (notably [Christodoulou and Klainerman 1990;Blue and Sterbenz 2006;Andersson and Blue 2009;Dafermos and Rodnianski 2009b;Luk 2010]) which in one form or the other are due to Morawetz [1962], or that relies on the exact stationarity of the spacetime (such as [Ching et al 1995;Donninger et al 2012] based on Fourier analytic methods). Here we follow the new physical-space approach to decay of [Dafermos and Rodnianski 2010], which only uses multipliers with weights in the radial variable.…”

Section: Introductioncontrasting

confidence: 66%

Untitled

2013

Anal. PDE

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See inside back cover or msp.org/apde for submission instructions.The subscription price for 2013 is US $160/year for the electronic version, and $310/year (+$35, if shipping outside the US) for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to MSP.Analysis & PDE (ISSN 1948(ISSN -206X electronic, 2157 We consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation g D 0 on the domain of outer communications of the Schwarzschild spacetime manifold .M n m ; g/ (where n 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux EOE . † / through a foliation of hypersurfaces † (terminating at future null infinity and to the future of the bifurcation sphere) decays, EOE . † / Ä CD= 2 , where C is a constant depending on n and m, and D < 1 is a suitable higher-order initial energy on † 0 ; moreover we improve the decay rate for the first-order energy to EOE@ t . † R / Ä CD ı = 4 2ı for any ı > 0, where † R denotes the hypersurface † truncated at an arbitrarily large fixed radius R < 1 provided the higher-order energy D ı on † 0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate j j † R Ä CD 0 ı = 3 2 ı . In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski (2010).

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Section: Introductioncontrasting

confidence: 66%

Untitled

2013

Anal. PDE

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“…In these cases, we were able to show that the space-time integral ũ L 4 (dtdρd 2 ω) is controlled by weighted H 1+ǫ norms. This built on previous work [3], in which the L ∞ norm was shown decay if weighted H 3 norms were bounded (and the initial data was small in the non-linear case). Similar results, with better decay estimates at the event horizon (an important part of the Schwarzschild manifold, corresponding to ρ → −∞) have also been proven [4].…”

Section: Introductionmentioning

confidence: 80%

“…Essentially, since M = R × S 2 is a 3-dimensional manifold, one expects the Hardy estimate from R 3 , u/|x| ≤ ∇u to hold; however, because there is no origin for R × S 2 , a little extra control is required. This result is taken from [3].…”

Section: Analysis Using the Methods Of Multipliersmentioning

confidence: 99%

A Space–Time Integral Estimate For A Large Data Semi-linear Wave Equation on the Schwarzschild Manifold

Blue

Soffer

2007

Lett Math Phys

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We consider the wave equation (−∂ 2 t +∂ 2 ρ −V −VL(−∆ S 2 ))u = f F ′ (|u| 2 )u with (t, ρ, θ, φ) in R×R×S 2 . The wave equation on a spherically symmetric manifold with a single closed geodesic surface or on the exterior of the Schwarzschild manifold can be reduced to this form. Using a smoothed Morawetz estimate which does not require a spherical harmonic decomposition, we show that there is decay in L 2 loc for initial data in the energy class, even if the initial data is large. This requires certain conditions on the potentials V , VL and f . We show that a key condition on the weight in the smoothed Morawetz estimate can be reduced to an ODE condition, which is verified numerically.We consider the following defocusing wave equation:Our goal is to show that, under conditions 1-11 on the potentials (see below),if the initial data (u 0 , u 1 ) has finite (but possibly large) energy. In fact, we show the stronger result that. Equation (1) describes several interesting geometric wave equations. On a spherically-symmetric, 3-dimensional, warped-product of R and S 2 , Riemannian manifold, the metric is ds 2 = dρ 2 + r(ρ) 2 dω 2 , and if r has a unique minimum, it corresponds to a closed geodesic surface. This is the first problem we consider, and, by translation, we may assume that the closed geodesic surface occurs at ρ = 0. The semi-linear wave equation (−∂ 2 t + ∆ g )ũ = |ũ| p−1ũ can be reduced to (1) by setting u = rũ, in which case V = r ′′ /r,. The conditions 1-11 on the potentials are, therefore, conditions on r. These conditions are not vacuous, since, for example r = 1 + ρ 2 generates a wave equation which satisfies these conditions. Similarly the wave equation on the exterior region of the 3 + 1-dimensional, Schwarzschild manifold also can be reduced to (1) if p = 3, and this is the second case we consider. Both cases are explained in more detail in [2].The purpose of this note is to show that there is decay even for large data (defocusing) semi-linear problems. In [2], we considered the linear equation ((1) in which the right-hand side is absent) in both the Riemannian and Schwarzschild cases, and the non-linear problem with small initial data in the Riemannian case. In these cases, we were able to show that the space-time integral ũ L 4 (dtdρd 2 ω) is controlled by weighted H 1+ǫ norms. This built on previous work [3], in which the L ∞ norm was shown decay if weighted H 3 norms were bounded (and the initial data was small in the non-linear case). Similar results, with better decay estimates at the event horizon (an important part of the Schwarzschild manifold, corresponding to ρ → −∞) have also been proven [4].We make the following assumptions on the potentials and non-linear terms:1

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“…We add that there have been many papers on decay of linear waves for Schwarzschild and Kerr black holes-see [2,9,18,19,22,23,28,29,46,47,49] and references given there. In that case the cosmological constant is 0 (unlike in the de Sitter case, where it is positive), and the methods of scattering theory are harder to apply because of an asymptotically Euclidean infinity.…”

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confidence: 99%

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

Dyatlov

2012

Ann. Henri Poincaré

132

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Abstract. We establish a Bohr-Sommerfeld type condition for quasinormal modes of a slowly rotating Kerr-de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a = 0 (Schwarzschild-de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.Quasi-normal modes (QNMs) of black holes are a topic of continued interest in theoretical physics: from the classical interpretation as ringdown of gravitational waves [11] to the recent investigations in the context of string theory [33]. The ringdown 1 plays a role in experimental projects aimed at the detection of gravitational waves, such as LIGO [1]. See [35] for an overview of the vast physics literature on the topic and [6,36,37,53] for some more recent developments.In this paper we consider the Kerr-de Sitter model of a rotating black hole and assume that the speed of rotation a is small; for a = 0, one gets the stationary Schwarzschild-de Sitter black hole. The de Sitter model corresponds to assuming that the cosmological constant Λ is positive, which is consistent with the current Lambda-CDM standard model of cosmology.

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Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space

Cited by 125 publications

References 5 publications

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A Space–Time Integral Estimate For A Large Data Semi-linear Wave Equation on the Schwarzschild Manifold

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

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