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

By P. Blue, Jacob Sterbenz. 2005

155 citations recorded by the source corpus.

AbstractWe provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzschild background. Our estimate implies that such solutions have asymptotic behavior $$|\phi| = O\left(r^{-1}\big|t-|r^*|\big|^{-\frac{1}{2}}\right)$$ as long as the source term is bounded in the norm $$(1-\frac{2M}{r})^{-1}\cdot(1 + t + |r^*|)^{-1}L^1\big(H^3_\Omega(r^2dr^* d\omega)\big)$$. In particular this gives scattering at small amplitudes for non-linear scalar fields of the form $$\square_{g}\varphi = \lambda |\varphi |^{p}\varphi $$ for all 2 < p.

Subfields: Control and Systems Engineering, Mathematical Physics, Rheumatology

View Paper: https://www.semanticscholar.org/paper/76e6b386d268e4b4589b1b8279d87f6cd349c378

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Citations in the corpus

Strichartz Estimates on Schwarzschild Black Hole Backgrounds (2008, methodology): “The approach of [25], [2], [7] and [14] is to write the equation using the Regge-Wheeler tortoise coordinate and to expand in spherical harmonics.”
Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009, methodology): “Following the introduction of a Morawetz vector field and of the equivalent of an almost conformal vector field to the Schwarzschild spacetime [34], decay estimates for the wave equation were proven [9], proven with a weaker decay rate but less regularity loss [8], and proven separately with a stron”
The red‐shift effect and radiation decay on black hole spacetimes (2005, background): “ore generalKerrfamily, even uniform boundedness remains an open problem (see however [20]). We should also note that, independently of us, a variant of the problem considered here is being studied by [3]. A statement of Theorem 1.1 in local coordinates will be given later. This, in particular, will ”
The evolution problem in general relativity (2008, background): “A result yielding similar decay away from the horizon (but weaker decay along the horizon) was proven independently in a nice paper of Blue and Sterbenz [19].”
Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole (2010, background): “The separation of variables techniques and the related symmetries have been used in many papers, including [2], [9], [11], [15, 16], [19, 20], [31], [37]; however, these mostly consider the case of zero cosmological constant, where other difficulties occur at zero energy and a global meromorphic con”
Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I (2011, background): “For Schwarzschild, such estimates appeared in [7, 8, 10, 21, 23] and for Reissner-Nordström in [9].”
Lectures on black holes and linear waves (2008, background): “Motivated by [65], the authors of [22] have since given a different argument [23] to obtain just the pointwise estimate (32) on the horizon, exploiting the “good” term in K near the horizon.”
Local decay of waves on asymptotically flat stationary space-times (2009, methodology): “Uniform pointwise t−1 decay estimates were obtained by Blue-Sterbenz [5] and also Dafermos-Rodnianski [16]; the bounds in the latter paper are stronger in that they extend uniformly up to the event horizon.”
A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds (2008, methodology): “Both [12] and [6] control trapping effects with the help of vector field multipliers which must be carefully chosen for each spherical harmonic separately.”