On Mañé's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem

In this paper, we introduce a nonlinear system of partial differential equations, the magnetic two-component Hunter-Saxton system (M2HS). This system is formulated as a magnetic geodesic equation on an infinite-dimensional Lie group equipped with a right-invariant metric, the $\dot{H}^1$ -metric, which is closely related to the infinite-dimensional Fisher-Rao metric, and the derivative of an infinite-dimensional contact-type form as the magnetic field. We define Mañé's critical value for exact magnetic systems on Hilbert manifolds in full generality and compute it explicitly for the (M2HS). Moreover, we establish an infinite-dimensional Hopf-Rinow theorem for this magnetic system, where Mañé's critical value serves as the threshold beyond which the Hopf-Rinow theorem no longer holds. This geometric framework enables us to thoroughly analyze the blow-up behavior of solutions to the (M2HS). Using this insight, we extend solutions beyond blow-up by introducing and proving the existence of global conservative weak solutions. This extension is facilitated by extending the Madelung transform from an isometry into a magnetomorphism, embedding the magnetic system into a magnetic system on an infinite-dimensional sphere equipped with the derivative of the standard contact form as the magnetic field. Crucially, this setup can always be reduced, via a dynamical reduction theorem, to a totally magnetic three-sphere, providing a deeper understanding of the underlying dynamics.