THE THREE-DIMENSIONAL STOCHASTIC ZAKHAROV SYSTEM

• 成果类型: Article
• 署名作者: Herr, Sebastian; Roeckner, Michael; Spitz, Martin; Zhang, Deng
• 署名单位: University of Bielefeld; Shanghai Jiao Tong University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1710
• 发表日期: 2025
• 页码: 848-905
• 关键词: nonlinear schrodinger-equations; global well-posedness; blow-up; SCATTERING; energy; strichartz; DYNAMICS; limit; MAPS

摘要

We study the three-dimensional stochastic Zakharov system in the energy space, where the Schr & ouml;dinger 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 & ouml;dinger 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 nonconservative noise. The key point of its proof is an estimate in Strichartz spaces for solutions of a Schr & ouml;dinger type equation with a nonlocal potential involving the free wave.