Global Cauchy problem of 2D generalized magnetohydrodynamic equations

In this paper we prove the global-in-time existence of smooth solutions of the 2D generalized magnetohydrodynamic equations with zero viscosity, when the power of the Laplacian operator −Δ in the dissipation term of the induction equation is between 1 and 2. We also show a regularity criterion on the direction of the magnetic field b0:=b|b| when β=1.