Weak solutions for two-dimensional magnetohydrodynamics equations with partial dissipation and magnetic diffusion

In this article, we study the existence of weak solutions to two-dimensional incompressible magnetohydrodynamics (MHD) equations. For the case with only magnetic diffusion, the global existence of solutions in $L^{\mathrm{\infty }}(0,\mathrm{\infty };H^1)$ was established by Kozono, whereas the existence of $L^{\mathrm{\infty }}(0,\mathrm{\infty };H^1)$ solutions for the case with only dissipation is totally unknown. The purpose here is to consider the mixed case, that is, a system with partial dissipation and partial magnetic diffusion. We show that there exists a unique local solution with initial data in $H^1$ space.