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

TL;DR

This paper proves that for the three-dimensional Zakharov system, randomized final states lead to unique scattering solutions in the energy space, utilizing randomization techniques and advanced estimates.

Contribution

It introduces a novel randomization approach for final states in the Zakharov system, establishing almost sure scattering without size restrictions or symmetry assumptions.

Findings

Almost sure existence of scattering solutions for randomized final states

Use of time-weighted norms and generalized Strichartz estimates

No restrictions on initial data size or symmetry

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.