Randomized final-state problem for the Zakharov system in dimension three

Abstract

We consider the final-state problem for the Zakharov system in the energy space in three space dimensions. For (u + , v + ) ∈ H 1 × L 2 without any size restriction, symmetry assumption or additional angular regularity, we perform a physical-space randomization on u + and an angular randomization on v + yielding random final states (u ω + , v ω + ). We obtain that for almost every ω, there is a unique solution of the Zakharov system scattering to the final state (u ω + , v ω + ). The key ingredient in the proof is the use of time-weighted norms and generalized Strichartz estimates which are accessible due to the randomization.