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2023 Three-dimensional magnetohydrodynamics system forced by space-time white noise
Kazuo Yamazaki
Author Affiliations +
Electron. J. Probab. 28: 1-66 (2023). DOI: 10.1214/23-EJP929


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].


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).


Download Citation

Kazuo Yamazaki. "Three-dimensional magnetohydrodynamics system forced by space-time white noise." Electron. J. Probab. 28 1 - 66, 2023.


Received: 8 April 2022; Accepted: 2 March 2023; Published: 2023
First available in Project Euclid: 6 March 2023

Digital Object Identifier: 10.1214/23-EJP929

Primary: 35B65 , 35Q85 , 35R60

Keywords: Gaussian hypercontractivity , magnetohydrodynamics system , Paracontrolled distributions , renormalization , Wick products

Vol.28 • 2023
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