A vector field method for non-trapping, radiating spacetimes

作者：Oliver Jesus^*

来源：Journal of Hyperbolic Differential Equations, 2016, 13(4): 735-790.

DOI：10.1142/S021989161650020X

• Vectorfield method
• local energy decay
• radiating spacetime
• conformal energy
• wave equation

We study the global decay properties of solutions to the linear wave equation in 1+3 dimensions on time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and a local energy decay estimate, we prove that sufficiently regular solutions to this equation have bounded conformal energy. As an application we also show a conformal energy estimate with vector fields applied to the solution as well as a global L-infinity decay bound in terms of a weighted norm on initial data. For solutions to the wave equation in these dynamical backgrounds, our results reduce the problem of establishing the classical pointwise decay rate t(-3/2) in the interior and t(-1) along outgoing null cones to simply proving that local energy decay holds.

• 出版日期2016-12