We study the incompressible fractional viscous–resistive magnetohydrodynamic system on Rn with fractional diffusion (−Δ)α, where α∈(1/2,1], and with positive viscosity and resistivity coefficients μ,ν>0. The problem is treated at the scale-invariant regularity sc=np+1−2α. For small divergence-free initial data in the critical Triebel–Lizorkin–Lorentz space F˙p,rsc,q, we construct a unique global mild solution. The main contribution is the use of the single-norm time–frequency space mm′F˙p,rsc,q, built on Meyer wavelets and the parabolic gauge t22αj. This space keeps the critical spatial size, the short-time behavior, and the high-frequency decay in one norm. By using a Gevrey-weighted Duhamel formulation, we prove boundedness of the corresponding fractional heat propagators and establish the bilinear paraproduct estimate required for the fixed-point argument. Consequently, e(t(−Δ)α)γ(u,b)∈mm′F˙p,rsc,q2n for some γ>0 depending on the parameters. This gives a Gevrey-type spatial smoothing effect, which is stronger than ordinary analyticity in the adopted scale. The restriction α>12 enters through the factor 2j(1−2α), which supplies the high-frequency gain needed to close the critical bilinear estimates; in this sense it is sharp for the present method. The classical viscous–resistive case is recovered when α=1.
Xie et al. (Sat,) studied this question.