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 - 2 α ( 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 2 , 1 s ( s ≥ n 2 + 1 ) .