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 ω, 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.

Keywords:

• Almost sure scattering
• final-state problem
• randomized data
• Zakharov system

Acknowledgment

I want to thank Sebastian Herr for invaluable advice on this project, instructive discussions, and helpful feedback. I also thank the anonymous referees for their useful comments.

Additional information

Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1283/2 2021 – 317210226.