Existence Of Stationary Solutions For A Class Of Compressible Magnetohydrodynamics Equations

The magnetohydrodynamics equations(MHD),which describe the motion of conductive fluid in magnetic field,is one of hot issues in the field of partial differential equations.This paper is intend to prove the existence of weak solutions for the compressible steady MHD equations with Coulomb force and give the L2 decay estimates for the strong solutions of the incompressible MHD equations with nonlinear damping terms.On the one hand,for the steady equation of compressible magnetohydrodynamics with Coulomb force,we construct a suitable approximation system,apply the Leray-Schauder fixed point theorem to obtain the existence of approximation solutions,and prove the existence of the weak solution to the steady equation when the specific heat ratio γ>2 with the help of the weak convergence method[36].The key point is get the uniform bound of approximation solutions.On the other hand,for the incompressible magnetohydrodynamics equation with nonlinear damping term,the L2 decay estimates for the strong solutions is obtained when the damping index 3/2≤α≤2 by using the energy method and the Poincare inequality.