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

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(-(1-1/p))), u(3)(t) decays at the rate of O (t(-3/2(1-1/p))) and B(t) decays at the rate of O (t(-3/2(1-1/p)-1/2)). Furthermore, we give the asymptotic expansion of solutions. We prove that the leading term of u(h)(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). For the 3D MHD system with full dissipation and horizontal magnetic diffusion, it is shown that B-h(t) decays at the rate of O(t(-(1-1/p))), B-3(t) decays at the rate of O(t(-3/2(1-1/p))), while u(t) decays at the rate of O(t(-9/8(1-1/p)-1/2) log(2+ t)). Moreover, we give the asymptotic expansion of solutions. We conclude that the leading term of B-h(t) is a combination of linear solution and two integrals from nonlinear coupling effects between velocity field and magnetic field, while for B-3(t) the leading term is given by only the linear solution without the influence of velocity field. For higher order asymptotic expansion of solutions, we show that the velocity field does affect the asymptotic expansion of B-3(t). Moreover, the nonlinear integrals of magnetic field turn out to be the second leading terms of higher order asymptotic expansion of u(t).