Abstract
We consider the three-dimensional magnetohydrodynamics system forced by noise that is white in both time and space. Its complexity due to four non-linear terms makes its analysis very intricate. Nevertheless, taking advantage of its structure and adapting the theory of paracontrolled distributions from [30], we prove its local well-posedness. A first challenge is to find an appropriate paracontrolled ansatz which must consist of both the velocity and the magnetic fields. Second challenge is that for some non-linear terms, renormalizations cannot be achieved individually; we overcome this obstacle by strategically coupling certain terms together rather than separately. Our proof is also inspired by the work of [70].
Acknowledgments
The author expresses deep gratitude to Prof. Carl Mueller and Prof. Marco Romito for valuable discussions. He also thanks Prof. Jared Whitehead for suggesting references [2, 28, 45, 58] on the Boussinesq system. Moreover, the author expresses deep gratitude to the anonymous referee and the editor for the valuable suggestions and comments that have improved this manuscript significantly. This work was supported by a grant from the Simons Foundation (962572, KY).
Citation
Kazuo Yamazaki. "Three-dimensional magnetohydrodynamics system forced by space-time white noise." Electron. J. Probab. 28 1 - 66, 2023. https://doi.org/10.1214/23-EJP929
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