Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

In this paper, we investigate the n-dimensional incompressible magnetohydrodynamic (MHD) equations with fractional dissipation and magnetic diffusion. Firstly, employing energy methods, we demonstrate that if the initial data is sufficiently small in H^s(ℝ^n) with s=1+n/2-2α (0<α <1) , then the system possesses a global solution. In order to establish the uniqueness, we enhance the regularity of the initial data and prove that if (u_0,b_0) is small in H^s(ℝ^n) with s=1+n/2-α (0<α <1) , then the system admits a unique global solution. Secondly, by applying frequency decomposition, we obtain ‖ u,b‖ _L^2→ 0, t→∞ . Assuming in addition that the initial data u_0,b_0∈ L^p(1≤ p<2) , we establish optimal decay estimates for the solutions and their higher order derivatives by employing a more refined frequency decomposition approach. In the case α = 0 , the system corresponds to a damped MHD equations, which have been previously investigated in [34]. Our results improve ones in [34] by extending the solution space from H^s(s>n/2+1) to B^s_2,1(s≥n/2+1) .