Global existence and uniqueness of weak solutions of a Stokes-Magneto system with fractional diffusions

We consider a Stokes-Magneto system in Rd (d≥2) with fractional diffusions Λ2αu and Λ2βb for the velocity u magnetic field b, respectively. Here α,β are positive constants Λs=(−Δ)s/2 is Laplacian of order s. establish global existence weak solutions any initial data L2 when α, β satisfy 1/2<α<(d+1)/2, β>0, min⁡{α+β,2α+β−1}>d/2. It also shown that unique if β≥1 min⁡{α+β,2α+β−1}≥d/2+1, addition.