Local existence and uniqueness of classical solutions for the ideal MHD equations in 2D exterior domains

In this paper, we consider the two-dimensional incompressible ideal MHD equations in the exterior of a rigid obstacle. We establish local existence and uniqueness of solutions for initial data in \(H^s\) $H^s$ ( \(s \ge 3\) $s\ge 3$ ). The proof relays heavily on the finding that, in the \(L^2\) $L^2$ setting, an irrotational field is always the gradient of a potential, despite the fact that the domain is not simply connected.