On scattering and profile decomposition for critical nonlinear waves outside weakly trapping obstacles

TL;DR

This paper establishes scattering for the energy-critical nonlinear wave equation outside convex obstacles in 3D, using profile decompositions and Strichartz estimates, even with trapped trajectories.

Contribution

It proves the first large data scattering result for such equations with obstacles and trapped trajectories, generalizing to other geometries.

Findings

Proves scattering for the nonlinear wave equation outside convex obstacles.

Establishes linear and nonlinear profile decompositions in infinite time.

Shows scattering under the condition that the only compact-flow solution is trivial.

Abstract

We prove scattering for the defocusing energy-critical non-linear wave equation with Dirichlet boundary conditions outside two strictly convex obstacles in dimension three. This is the first large data scattering result for such an equation in the presence of trapped trajectories. Our result is in fact more general and can be used as a black box in other geometries. More precisely, under the assumptions that the corresponding linear wave equation satisfies global Strichartz estimates, that the domain is weakly non-trapping and that trajectories do not reconcentrate, we show linear and nonlinear profile decompositions in infinite time. This implies scattering under the rigidity assumption that the only compact-flow solution is the trivial one.