On the global regularity of two-dimensional generalized magnetohydrodynamics system

We study the two-dimensional generalized magnetohydrodynamics system with dissipation and diffusion in terms of fractional Laplacians. In particular, we show that in case the diffusion term has the power $\beta = 1$, in contrast to the previous result of $\alpha \geq \frac{1}{2}$, we show that $\alpha > \frac{1}{3}$ suffices in order for the solution pair to remain smooth for all time.