Large time behavior of the solutions to 3D incompressible MHD system with horizontal dissipation or horizontal magnetic diffusion

Abstract

In this paper, we consider the asymptotic behavior of global solutions to 3D anisotropic incompressible MHD systems. For the 3D MHD system with horizontal dissipation and full magnetic diffusion, it is shown that $\uh(t)$ decays at the rate of $O(t^{-(1-\f{1}{p})})$, $u_3(t)$ decays at the rate of $O(t^{-\f{3}{2}(1-\f{1}{p})})$ and $B(t)$ decays at the rate of $O(t^{-\f{3}{2}(1-\f{1}{p})-\f{1}{2}})$. Furthermore, we give the asymptotic expansion of solutions. We prove that the leading term of $\uh(t)$ is a combination of linear solution and two integrals from nonlinear coupling effects, while for $u_3(t)$ the leading term is given by only the linear solution without the influence of magnetic field. Though the dissipation of velocity is weak, we show that the full magnetic diffusion is robust enough to keep the asymptotic expansion of magnetic field basically expected. However, the magnetic field turns out to affect the higher order asymptotic expansions of $u_3(t)$. The obtained results reveal the crucial role played by the magnetic field in the asymptotic analysis of anisotropic incompressible MHD equations and are expected to enhance the understanding of incompressible MHD equations from the viewpoint of mathematics.