The energy-critical stochastic Zakharov system

TL;DR

This paper investigates the stochastic Zakharov system in four dimensions, establishing local and global well-posedness results, and demonstrating a regularization by noise effect that enhances solution existence.

Contribution

It introduces a new analytical framework for the energy-critical stochastic Zakharov system, proving well-posedness, global existence under certain conditions, and a noise-induced regularization phenomenon.

Findings

Local well-posedness in energy space $H^1\times L^2$

Global existence for solutions below ground state energy

Regularization by noise leading to increased probability of global solutions

Abstract

This work is devoted to the stochastic Zakharov system in dimension four, which is the energy-critical dimension. First, we prove local well-posedness in the energy space $H^1\times L^2$ up to the maximal existence time and a blow-up alternative. Second, we prove that for large data solutions exist globally as long as energy and wave mass are below the ground state threshold. Third, we prove a regularization by noise phenomenon: the probability of global existence and scattering goes to one if the strength of the (non-conservative) noise goes to infinity. The proof is based on the refined rescaling approach and a new functional framework, where both Fourier restriction and local smoothing norms are used as well as a (uniform) double endpoint Strichartz and local smoothing inequality for the Schr\"odinger equation with certain rough and time dependent lower order perturbations.