TY - JOUR
T1 - A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds
AU - Dafermos, Mihalis
AU - Rodnianski, Igor
PY - 2011/9
Y1 - 2011/9
N2 - We consider Kerr spacetimes with parameters a and M such that {pipe}a{pipe}«M, Kerr-Newman spacetimes with parameters {pipe}Q{pipe}«M, {pipe}a{pipe}«M, and more generally, stationary axisymmetric black hole exterior spacetimes (M,g) which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □gψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Στ, then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Στ is bounded by C times the initial energy flux through Σ0. This energy flux is positive definite and does not degenerate at the horizon, i. e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ0. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.
AB - We consider Kerr spacetimes with parameters a and M such that {pipe}a{pipe}«M, Kerr-Newman spacetimes with parameters {pipe}Q{pipe}«M, {pipe}a{pipe}«M, and more generally, stationary axisymmetric black hole exterior spacetimes (M,g) which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □gψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Στ, then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Στ is bounded by C times the initial energy flux through Σ0. This energy flux is positive definite and does not degenerate at the horizon, i. e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ0. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.
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U2 - 10.1007/s00222-010-0309-0
DO - 10.1007/s00222-010-0309-0
M3 - Article
SN - 0020-9910
VL - 185
SP - 467
EP - 559
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 3
ER -