Decay of solutions to the Maxwell equation on the Schwarzschild background

Abstract

A new Morawetz or integrated local energy decay estimate for Maxwell test fields on the exterior of a Schwarzschild black hole spacetime is proved. The proof makes use of a new superenergy tensor $H_{ab}$ defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current $H_{ab}({\partial }_t{)}^b$. The tensor $H_{ab}$ vanishes for the static Coulomb field, and the Morawetz estimate proved here therefore yields integrated decay for the Maxwell field to the Coulomb solution on the Schwarzschild exterior.

15 pages, updated references

References in corpus (5)

• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• Errata for ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'', ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'', and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation''
• Spin geometry and conservation laws in the Kerr spacetime
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• Spin geometry and conservation laws in the Kerr spacetime
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• Price's law for spin fields on a Schwarzschild background
• Almost Price's law in Schwarzschild and decay estimates in Kerr for Maxwell field
• The spin $\pm$1 Teukolsky equations and the Maxwell system on Schwarzschild
• Decay of Maxwell Fields on Reissner-Nordstrøm-de Sitter Black Holes
• Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation
• The decay of the $SU(2)$ Yang-Mills fields on the Schwarzschild black hole for spherically symmetric small energy initial data