On the Vanishing Viscosity and Magnetic Diffusion Limit for Incompressible Magnetohydrodynamics

The article focuses on the inviscid and non-resistive limit of the incompressible magnetohydrodynamics (MHD) equations in a bounded domain. We derive sufficient conditions for the limit of Leray-Hopf solutions of the MHD system, which converges to the weak solutions of the ideal MHD system when $\mu,\nu\rightarrow0$, under the assumptions of uniform interior regularity and uniform equi-continuity at the boundary. Therefore, we establish an analogue of the Kato-type criterion to show that the energy dissipation rate is independent of the coefficients of viscosity and resistivity. Moreover, to analyze the different decay rates for the viscosity and resistivity, we obtain the limits of infinite and vanishing magnetic Prandtl number ($Pm=\frac{\mu}{\nu}$) for the incompressible MHD system. To the best of the authors' knowledge, this is the first rigorous result from a mathematical viewpoint on the limit of the magnetic Prandtl number.