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 $(u_+, v_+) \in H^1 \times L^2$ 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 $(u_+^\omega, v_+^\omega)$. We obtain that for almost every $\omega$, there is a unique solution of the Zakharov system scattering to the final state $(u_+^\omega, v_+^\omega)$. The key ingredient in the proof is the use of time-weighted norms and generalized Strichartz estimates which are accessible due to the randomization.