Analysis Seminar

In this joint talk, we present an analysis for deriving precise asymptotics within the full forward causal cone and also throughout spacetime for solutions to non-linear wave equations on stationary asymptotically flat (3+1)-dimensional spacetimes including black holes. Our approach, inspired by A. Vasy's recent work on low-energy analysis and Sobolev estimates, along with P. Hintz's application to linear waves via resolvent expansions, is extended to a broader context. We successfully apply these methods to non-linear hyperbolic equations; for example, we conclusively establish the precise asymptotics for non-linear wave equations satisfying the null condition. This covers cubic power non-linear wave equations, where lower power nonlinearities present the greatest challenges in asymptotic analysis, since they are farthest away from linear wave equations. We combine those spectral methods with integrated local energy decay (i.e., Morawetz estimates) and pointwise decay for non-linear waves, and the development of second-order radiation field expansions which we expect can be carried out to any finite order.

Thus, our approach is expected to provide an arbitrarily precise (i.e., asymptotic expansion to all finite orders) characterization of the late-time behavior of global solutions to non-linear wave equations, and moreover we produce asymptotics in all possible asymptotic regimes of the spacetime, including all joint large-time, large-radii regimes.