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

Abstract

In this article, the two-dimensional magneto-hydrodynamic (MHD) equations are considered with only magnetic diffusion. Here the magnetic diffusion is given by \({\mathfrak D}\) a Fourier multiplier whose symbol m is given by \(m(\xi )=|\xi |^2\log (e+|\xi |^2)^\beta \). We prove that there exists an unique global solution in \(H^s(\mathbb {R}^2)\) with \(s>2\) for these equations when \(\beta >1\). This result improves the previous works which require that \(m(\xi )=|\xi |^{2\beta }\) with \(\beta >1\) 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 \(m(\xi )=|\xi |^2\).