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

Abstract

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 $H^s(\mathbb {R}^n)$ with $s=1+\frac{n}{2}-2\alpha ~(0<\alpha <1)$, 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 $H^s(\mathbb {R}^n)$ with $s=1+\frac{n}{2}-\alpha ~(0<\alpha <1)$, then the system admits a unique global solution. Secondly, by applying frequency decomposition, we obtain $\Vert u,b\Vert _{L^2}\rightarrow 0,~t\rightarrow \infty$. Assuming in addition that the initial data $u_0,b_0\in L^p(1\le p<2)$, we establish optimal decay estimates for the solutions and their higher order derivatives by employing a more refined frequency decomposition approach. In the case $\alpha = 0$, 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 $H^s(s>\frac{n}{2}+1)$ to $B^s_{2,1}(s\ge \frac{n}{2}+1)$.