Existence Of Solutions And Parameter Estimation For Stochastic Incompressible Magnetohydrodynamic Equations

In this paper,we consider a three-dimensional stochastic incompressible magnetohydrodynamic equation with an external force,and study the existence of the solution and the problem of parameter estimation.Firstly,we assume that the noise and initial data in the equation are smooth with respect to the space variables,and apply appropriate cut-off function to deal with the nonlinear convection term.The equation is transformed into determination system with a new unknown external force.By constructing a suitable mapping and a functional space,the existence and uniqueness of solutions to deterministic equations are obtained in a small time interval by using the contraction mapping theorem under some assumption of external force.By establishing a uniform energy estimate,the solution will be extended to a large time interval,and the continuous dependence of the solution on the initial value is proved.Secondly,the existence and uniqueness of the solution to the stochastic modified equation driven by additive noise are obtained by using polishing technique,energy estimation and stopping time.Furthermore,the existence and uniqueness of solutions for the modified equation driven by multiplicative noise are obtained by using the results of the stochastic modified equation with additive noise.Thirdly,the cut-off function is removed,and the local existence and uniqueness of strong solutions for the original equation in probability sense are obtained by using Chebyshev's inequality and the results of the modified equation driven by multiplicative noise.Furthermore,the existence and uniqueness of global strong solutions in the sense of probability are proved by stopping time and energy estimates when the initial value and external force are small enough.Finally,combining the idea of numerical difference with least squares principal,the parameter estimation of one-dimensional stochastic MHD equations driven by the white noise are considered respectively.The feasibility of this parameter estimation method is verified by numerical simulation of the specific equation.The results show that the step size and noise factor have some influence on the parameter estimation of the equation.The smaller the step size and noise coefficient,the closer the estimated parameters are to the true values.When the appropriate step size and noise factor are taken,the parameters of the equation can be estimated more accurately by the parameter estimation method in this paper.