Logarithmically improved blow-up criterion for smooth solutions to the Leray-$\alpha $-magnetohydrodynamic equations

(1.1)  ∂tv + (u · ∇)v −∆v +∇π + 2∇|B| 2 = (B · ∇)B, ∂tB + (u · ∇)B − (B · ∇)v −∆B = 0, v = (1− α2∆)u, α > 0, div u = div v = divB = 0, (v,B)|t=0 = (v0, B0),div u0 = div v0 = divB0 = 0 in R3, where v : the fluid velocity field, u : “the filtered” fluid velocity, B : the magnetic field and π : the pressure, are the unknowns; α is the lengthscale parameter that represents the width of the filter. Note that the magnetic field is not regularized. It has lately received significant attention in mathematical fluid dynamics due to its connection to three-dimensional incompressible flows. When α→ 0, the model (1.1) reduce to the following MHD equations: