The three dimensional stochastic Zakharov system

TL;DR

This paper investigates the well-posedness and blow-up behavior of the three-dimensional stochastic Zakharov system, demonstrating that noise can prevent finite-time blowup with high probability.

Contribution

It establishes well-posedness, blow-up criteria, and a noise-induced regularization effect for the stochastic Zakharov system in three dimensions.

Findings

Solutions exist as long as they stay below the ground state.

Finite-time blowup can be prevented by adding sufficiently large noise.

The proof employs refined rescaling, normal form method, and local smoothing estimates.

Abstract

We study the three dimensional stochastic Zakharov system in the energy space, where the Schr\"odinger equation is driven by linear multiplicative noise and the wave equation is driven by additive noise. We prove the well-posedness of the system up to the maximal existence time and provide a blow-up alternative. We further show that the solution exists at least as long as it remains below the ground state. Two main ingredients of our proof are refined rescaling transformations and the normal form method. Moreover, in contrast to the deterministic setting, our functional framework also incorporates the local smoothing estimate for the Schr\"odinger equation in order to control lower order perturbations arising from the noise. Finally, we prove a regularization by noise result which states that finite time blowup before any given time can be prevented with high probability by adding sufficiently large non-conservative noise. The key point of its proof is an estimate in Strichartz spaces for solutions of a Schr\"odinger type equation with a nonlocal potential involving the free wave.