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

Yanbin Tang,Zhaoyang Qiu

doi:10.3390/e22010042

Abstract

In this paper, we consider the existence of local smooth solution to stochastic magneto-hydrodynamic equations without diffusion forced by additive noise in . 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.

Highlights

The magneto-hydrodynamic (MHD) equations have a wide range of applications in geophysics, astrophysics, and plasma physics [1,2,3,4,5,6,7,8,9,10]
Kim [18] established the existence of local smooth solution to 3D stochastic Euler equation forced by additive noise
No one has addressed the existence of the local smooth solution to the stochastic MHD equations without diffusion driven by additive noise when the initial data belongs to H α (R3 ), α > 52

Summary

Introduction

The magneto-hydrodynamic (MHD) equations have a wide range of applications in geophysics, astrophysics, and plasma physics [1,2,3,4,5,6,7,8,9,10]. Kim [18] established the existence of local smooth solution to 3D stochastic Euler equation forced by additive noise. No one has addressed the existence of the local smooth solution to the stochastic MHD equations without diffusion driven by additive noise when the initial data belongs to H α (R3 ), α > 52. By the Chebyshev inequality and energy estimates, the probability of existence can be made arbitrarily close to one

Construction of Approximate Solution

Existence and Uniqueness of the Local Smooth Solution