Three-dimensional magnetohydrodynamics system forced by space-time white noise

Kazuo Yamazaki

Published 2019-10-10Version 1

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 \cite{GIP15}, 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 employing a technique which may be appropriately called a coupled renormalization. This technique of coupled renormalizations seems to be new and is expected to be crucial for any other systems of non-linearly coupled differential equations such as the Boussinesq system. Our proof is also inspired by the work of \cite{ZZ15}. To the best of the author's knowledge, this is the first result of a well-posedness of a non-linearly coupled system of equations forced by a space-time white noise.