Existence and Uniqueness of the Local Smooth Solution to 3D Stochastic MHD Equations without Diffusion

Abstract

In this paper, we consider the existence of local smooth solution to stochastic magneto-hydrodynamic equations without diffusion forced by additive noise in R 3 . We first transform the system into a random system via a simple change of variable and borrow the result obtained for classical magneto-hydrodynamic equations, then we show that this random transformed system is measurable with respect to the stochastic element. Finally we extend the solution to the maximality solution. Due to the coupled construction of this system, we need more elaborate and complicated estimates with respect to stochastic Euler equation.

Keywords

• stochastic magneto-hydrodynamic equations
• smooth solution
• the kato method
• stopping time
• progressive measurability