On the global regularity issue of the two-dimensional magnetohydrodynamics system with magnetic diffusion weaker than a laplacian

In this manuscript, we discuss the recent developments in the research direction concerning whether the solution to the two-dimensional magnetohydrodynamics system with certain velocity dissipation and magnetic diffusion strengths preserves the high regularity of a given initial data for all time or exhibits a blowup in finite time. In particular, we address an open problem in case the magnetic diffusion is weaker than a full Laplacian. In short, we consider this system with both velocity dissipation and magnetic diffusion strength measured in terms of fractional Laplacians with certain powers, and point out the following gap in the results among the current literature. In case the power of the fractional Laplacian representing the magnetic diffusion is one, the global well-posedness follows as long as the velocity dissipation is present regardless of how weak its strength may be, and hence the sum of the two powers need to only be more than one ([Monatsh. Math. 175 (2014), no. 1, 127–131], [Second proof of the global regularity of the two-dimensional MHD system with full diffusion and arbitrary weak dissipation, Methods Appl. Anal., to appear]). On the other hand, once the power of the fractional Laplacian representing the magnetic diffusion drops below one, in order to ensure the system’s global well-posedness, the sum of the powers from the dissipation and diffusion must be equal to or more than two, improved only logarithmically ([Nonlinear Anal. 85 (2013), 43–51], [J. Math. Fluid Mech. 13 (2011), no. 2, 295–305], [Appl. Math. Lett. 29 (2014), 46–51]). We discuss this issue in more detail, explain its reasoning, and provide some basic regularity criteria to gain better insight to this difficult direction of research.