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 × L 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 (uω+, v ω +). We obtain that for almost every ω, there is a unique solution of the Zakharov system scattering to the final state (uω+, 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.

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