The three dimensional stochastic Zakharov system

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.