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

Overview & Synopsis

In this article, the two-dimensional magneto-hydrodynamic (MHD) equations are considered with only magnetic diffusion. Here the magnetic diffusion is given by ${\mathcal D}$ 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 $m(\xi) = |\xi|^2$.