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Decay and the conformal energy of waves around the Schwarzschild black hole

Wednesday 16th November 2005 - 16:00 to 17:00
INI Seminar Room 1

We provide decay estimates for solutions to the decoupled, inhomogeneous wave equation around a Schwarzschild black hole. In Euclidean space, the conformal charge is a conserved quantity which is used to prove the decay of the local energy. The analogue around a Schwarzschild black hole can grow because of trapping near the photon sphere at r=3M. The trapping terms can be controlled by a Mourre estimate. However, compared to the Euclidean case, this requires more angular differentiability and allows the local energy to decay more slowly. One refinement of this method reduces the loss of angular differentiability, and another recovers the Euclidean rate of decay for the local energy. From the faster decay result, solutions decay like |\phi|=O(r^{-1} |t-|r_*||^{-\frac{1}{2}}). This is sufficient to prove small-data global-well-posedness for certain non linear problems. The initial data can be general in the sense that it is not composed of finitely spherical harmonic modes.

Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons