Long Time Behavior For A Class Of 2D Generalized Magnetohydrodynamic Equations

Recent years, researchers mainly do a lot of works for the 2D generalized MHD equations. For MHD equations of research is still relatively weak. So it is necessary to conduct more thorough research to the 2D generalized MHD equation.In this paper,we consider the long time behavior of 2D generalized MHD equa-tion: where Ω (?) Rn is a bounded domain with a sufficiently smooth boundary (?)Ω,u is the velocity vector field,v is the magnetic vector field,u0(x),v0(x) are the initial velocity and magnetic fields,respectively,α,β>n/2 are the parameters,γ> 0,η> 0 are the kinematic viscosity and diffusivity constants,respectively. A=(-Δ)is a Laplacian operator.This paper mainly consists of the following three chapters to discuss the long time behavior of the 2D generalized MHD equation:In the first chapter,we mainly introduces the current situation and physical background of the 2D generalized MHD equations.In the second chapter,we mainly researches the existence and uniqueness of global solutions,the existence of the global attractor and the upper bound estima-tion of the 2D generalized MHD equation about Hausdorff dimension and fractal dimension.In the third chapter,we mainly researches the inertial manifold of the 2D gen-eralized MHD equation when α=β,γ=η.