Strichartz estimates on Schwarzschild black hole backgrounds

Abstract

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.

44 pages; typos fixed, minor modifications in several places

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Cited by in corpus (59)

• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I
• The linear stability of the Schwarzschild solution to gravitational perturbations
• A new physical-space approach to decay for the wave equation with applications to black hole spacetimes
• The black hole stability problem for linear scalar perturbations
• The global non-linear stability of the Kerr-de Sitter family of black holes
• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II
• A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds
• Analysis of linear waves near the Cauchy horizon of cosmological black holes
• A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta
• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• Improved decay for solutions to the linear wave equation on a Schwarzschild black hole
• Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations
• Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces
• Asymptotics of linear waves and resonances with applications to black holes
• Quasimodes and a Lower Bound on the Uniform Energy Decay Rate for Kerr-AdS Spacetimes
• The Strauss conjecture on Kerr black hole backgrounds
• Non-linear stability of the Kerr-Newman-de Sitter family of charged black holes
• Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime
• Decay of linear waves on higher dimensional Schwarzschild black holes
• Pointwise decay for the Maxwell field on black hole space-times
• Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation
• Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein-Gordon system
• The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge
• Long time existence for semilinear wave equations on asymptotically flat space-times
• Global-in-time Strichartz estimates on non-trapping asymptotically conic manifolds
• Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces
• Recent works on the Strauss conjecture
• A sharp version of Price's law for wave decay on asymptotically flat spacetimes
• The Strauss conjecture on asymptotically flat space-times
• Asymptotics for the wave equation on differential forms on Kerr-de Sitter space
• Linear Stability of Schwarzschild Spacetime: Decay of Metric Coefficients
• Sharp decay estimates for massless Dirac fields on a Schwarzschild background
• The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Stability and Instability Results
• Radiation fields on Schwarzschild spacetime
• A Vector Field Method for Radiating Black Hole Spacetimes
• Price's law for spin fields on a Schwarzschild background
• Strichartz estimates on Kerr black hole backgrounds
• A local energy estimate for wave equations on metrics asymptotically close to Kerr
• Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes
• Lifespan estimates for local solutions to the semilinear wave equation in Einstein-de Sitter spacetime
• Pointwise decay for the wave equation on nonstationary spacetimes
• On quasilinear Maxwell equations in two dimensions
• On pointwise decay of linear waves on a Schwarzschild black hole background
• On the global stability of the wave-map equation in Kerr spaces with small angular momentum
• Pointwise decay for semilinear wave equations on Kerr spacetimes
• Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping
• A vector field method on the distorted Fourier side and decay for wave equations with potentials
• Mathematical general relativity: a sampler
• Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities
• Localized energy estimates for wave equations on high dimensional Schwarzschild space-times
• Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime
• Linear stability of Schwarzschild spacetime subject to axial perturbations
• Nonlinear Wave Equations With Null Condition On Extremal Reissner-Nordström Spacetimes I: Spherical Symmetry
• Localized energy estimates on Myers-Perry space-times
• The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting
• The weak null condition on Kerr backgrounds
• Physical-space estimates for axisymmetric waves on extremal Kerr spacetime
• Integrated Local Energy Decay for Damped Magnetic Wave Equations on Stationary Space-Times
• Lifespan estimates for $2$-dimensional semilinear wave equations in asymptotically Euclidean exterior domains