Reconnection in 2D Inertial Magnetohydrodynamics

We consider two-fluid equations of inertial magnetohydrodynamics in two spatial dimensions. These equations can be recast as the evolution of two active scalars, with fluid-like interaction kernels. The equations come in two flavors, left-handed and right-handed. They are conservative and have global unique solutions for both smooth initial data and for initial data in Yudovich class $L^1({\mathbb{R}}^2)\cap L^{\mathrm{\infty }}({\mathbb{R}}^2)$. We prove that the inviscid limit holds in Yudovich class. We prove the persistence of $C^{2,\alpha }$ regularity of patch boundaries through merger events for the left-handed equations. We finally show numerical examples of magnetic reconnection, validating known rigorous results and uncovering new cases.