Global regularity for logarithmically critical 2D MHD equations with zero viscosity

In this article, the two-dimensional magneto-hydrodynamic (MHD) equations are considered with only magnetic diffusion. Here the magnetic diffusion is given by D a Fourier multiplier whose symbol m is given by m(ξ) = |ξ| log(e + |ξ|) . We prove that there exists an unique global solution in H(R) with s > 2 for these equations when β > 1. This result improves the previous works which require that m(ξ) = |ξ| with β > 1 and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely m(ξ) = |ξ|.