A Decay Estimate for a Wave Equation with Trapping and a Complex Potential

International Mathematics Research Notices (Impact Factor: 1.1). 07/2011; 2013(3). DOI: 10.1093/imrn/rnr237
Source: arXiv


In this brief note, we consider a wave equation that has both trapping and a
complex potential. For this problem, we prove a uniform bound on the energy and
a Morawetz (or integrated local energy decay) estimate. The equation is a model
problem for certain scalar equations appearing in the Maxwell and linearised
Einstein systems on the exterior of a rotating black hole.

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    ABSTRACT: We study the one-dimensional wave equation with an inverse power potential that equals $const.x^{-m}$ for large $|x|$ where $m$ is any positive integer greater than or equal to 3. We show that the solution decays pointwise like $t^{-m}$ for large $t$, which is consistent with existing mathematical and physical literature under slightly different assumptions (see e.g. Bizon, Chmaj, and Rostworowski, 2007; Donninger and Schlag, 2010; Schlag, 2007). Our results can be generalized to potentials consisting of a finite sum of inverse powers, the largest of which being $const.x^{-\alpha}$ where $\alpha>2$ is a real number, as well as potentials of the form $const.x^{-m}+O(x^{-m-\delta_1})$ with $\delta_1>3$.
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    ABSTRACT: We consider the Maxwell equation in the exterior of a very slowly rotating Kerr black hole. For this system, we prove the boundedness of a positive definite energy on each hypersurface of constant $t$. We also prove the convergence of each solution to a stationary Coulomb solution. We separate a general solution into the charged, Coulomb part and the uncharged part. Convergence to the Coulomb solutions follows from the fact that the uncharged part satisfies a Morawetz estimate, i.e. that a spatially localised energy density is integrable in time. For the unchanged part, we study both the full Maxwell equation and the Fackerell-Ipser equation for one component. To treat the Fackerell-Ipser equation, we use a Fourier transform in $t$. For the Fackerell-Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.


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