| Abstract: |
| We established existence and uniqueness results for stochastic Magnetohydrodynamics equations in probabilistic evolution spaces of Besov type. This is done first by establishing regularity results for the stochastic heat equation with additive noise in the space ${\cal L}_\Omega^\sigma{\cal L}_T^\infty(\dot{B}_{2,q}^\frac{s}{\sigma})\cap{\cal L}_\Omega^\sigma{\cal L}_T^\sigma(\dot{B}_{2,q}^\frac{s+2}{\sigma})$; where $s\in\mathbb{R},\sigma\in[2,\infty)$ and $q\in[2,\infty]$. The regularity result is used to prove a global and local in time existence and uniqueness of solution to the 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. |
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