Small-amplitude nonlinear waves on a black hole background

Mihalis Dafermos; Igor Rodnianski

doi:10.1016/j.matpur.2005.04.004

Small-amplitude nonlinear waves on a black hole background

Mihalis Dafermos, Igor Rodnianski

Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

Let G(x) be a C⁰ function such that G(x) ≤ K x ^p for x ≤ c, for constants K, c > 0. We consider spherically symmetric solutions of □_gφ = G(φ) where g is a Schwarzschild or more generally a Reissner-Nordström metric, and such that φ and ∇φ are compactly supported on a complete Cauchy surface. It is proven that for p > 4, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, φ ≤ C(max{ν, 1})^-1, where ν denotes an Eddington-Finkelstein advanced time coordinate.

Original language	English (US)
Pages (from-to)	1147-1172
Number of pages	26
Journal	Journal des Mathematiques Pures et Appliquees
Volume	84
Issue number	9
DOIs	https://doi.org/10.1016/j.matpur.2005.04.004
State	Published - Sep 2005

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

Keywords

Black hole
Non-linear wave equation
Reissner-Nordström
Schwarzschild

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10.1016/j.matpur.2005.04.004

Cite this

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title = "Small-amplitude nonlinear waves on a black hole background",

abstract = "Let G(x) be a C0 function such that G(x) ≤ K x p for x ≤ c, for constants K, c > 0. We consider spherically symmetric solutions of □gφ = G(φ) where g is a Schwarzschild or more generally a Reissner-Nordstr{\"o}m metric, and such that φ and ∇φ are compactly supported on a complete Cauchy surface. It is proven that for p > 4, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, φ ≤ C(max\{ν, 1\})-1, where ν denotes an Eddington-Finkelstein advanced time coordinate.",

keywords = "Black hole, Non-linear wave equation, Reissner-Nordstr{\"o}m, Schwarzschild",

author = "Mihalis Dafermos and Igor Rodnianski",

note = "Funding Information: M.D. is supported in part by NSF grant DMS-0302748. I.R. is supported in part by NSF grant DMS-0406627. Part of this research was conducted when I.R. was a long-term prize fellow of the Clay Mathematics Institute.",

year = "2005",

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doi = "10.1016/j.matpur.2005.04.004",

language = "English (US)",

volume = "84",

pages = "1147--1172",

journal = "Journal des Mathematiques Pures et Appliquees",

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publisher = "Elsevier Masson s.r.l.",

number = "9",

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T1 - Small-amplitude nonlinear waves on a black hole background

AU - Dafermos, Mihalis

AU - Rodnianski, Igor

N1 - Funding Information: M.D. is supported in part by NSF grant DMS-0302748. I.R. is supported in part by NSF grant DMS-0406627. Part of this research was conducted when I.R. was a long-term prize fellow of the Clay Mathematics Institute.

PY - 2005/9

Y1 - 2005/9

N2 - Let G(x) be a C0 function such that G(x) ≤ K x p for x ≤ c, for constants K, c > 0. We consider spherically symmetric solutions of □gφ = G(φ) where g is a Schwarzschild or more generally a Reissner-Nordström metric, and such that φ and ∇φ are compactly supported on a complete Cauchy surface. It is proven that for p > 4, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, φ ≤ C(max{ν, 1})-1, where ν denotes an Eddington-Finkelstein advanced time coordinate.

AB - Let G(x) be a C0 function such that G(x) ≤ K x p for x ≤ c, for constants K, c > 0. We consider spherically symmetric solutions of □gφ = G(φ) where g is a Schwarzschild or more generally a Reissner-Nordström metric, and such that φ and ∇φ are compactly supported on a complete Cauchy surface. It is proven that for p > 4, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, φ ≤ C(max{ν, 1})-1, where ν denotes an Eddington-Finkelstein advanced time coordinate.

KW - Black hole

KW - Non-linear wave equation

KW - Reissner-Nordström

KW - Schwarzschild

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DO - 10.1016/j.matpur.2005.04.004

M3 - Article

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VL - 84

SP - 1147

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JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

IS - 9

ER -

Small-amplitude nonlinear waves on a black hole background

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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