Harmonic analysis tools for stochastic magnetohydrodynamics equations in Besov spaces

Abstract

We establish a regularity result for stochastic heat equations in probabilistic evolution spaces of Besov type and we use it to prove a global in time existence and uniqueness of solution to a stochastic magnetohydrodynamics equation. The existence result holds with a positive probability which can be made arbitrarily close to one. The work is carried out by blending harmonic analysis tools such as Littlewood–Paley decomposition, Jean–Micheal Bony paradifferential calculus and stochastic calculus. The law of large numbers is a key tool in our investigation. Our global existence result is new in three-dimensional spaces.

Local and global strong solutions to the stochastic incompressible Navier-Stokes equations in critical Besov space

“…It is the particular case of Theorem 3.5 in [25], but the proof is more simple. For the special case of Proposition 6.5 inḂ s 2,r , we would refer the reader to [22].…”

“…We point out that our conditions of the noise f (t, u) is more general than the linear map of u. After suitable modifying, we cover the result of [22], which consider the MHD equations with the additive noise in Ḃ d 2 −1 2,r . However, when p > d, one may take the initial velocity in a Besov space with a negative index of regularity, so that a highly oscillating "large "initial velocity u ǫ 0 in (1.2) may give rise to a unique global solution.…”