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.

28 pages v2: typos corrected, uniqueness statement in Corollary 1.3 improved