Invariant measures and statistical solutions of the globally modified magnetohydrodynamics equations

Abstract

In this paper, we consider a three-dimensional nonlinear system consisting of the incompressible Navier-Stokes equations governing fluid motion coupled with the incompressible Maxwell's equations governing the magnetic field. This system is subject to a no-slip boundary condition for the (average) velocity and the perfectly conducting wall condition for the magnetic field. We prove some regularity properties of solutions of this system through an approximation procedure based on Galerkin approximation scheme. Asa consequence, we are able to derive some suitable uniform estimates which allow us to show smoothness of the global attractor. Finally, we discuss the relationship between global attractors, invariant measures, time-average measures, and statistical solutions of the three-dimensional system of globally modified magmnetohydrodynamics equations in the case of temporally independent forcing terms.