On Mañé’s Critical Value for the Two-Component Hunter–Saxton System and an Infinite-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, namely the Ḣ¹ H ˙ 1 -metric, which is closely related to the infinite-dimensional Fisher–Rao metric. The magnetic field is given by the exterior derivative of an infinite-dimensional contact-type form. 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 property fails. This geometric framework enables a detailed analysis of 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 achieved by extending the Madelung transform from an isometry to a magnetomorphism, thereby embedding the magnetic system into a magnetic system on an infinite-dimensional sphere equipped with the exterior derivative of the standard contact form as the magnetic field. Crucially, this setting can always be reduced, via a dynamical reduction theorem, to a totally magnetic three-sphere, providing deeper insight into the underlying dynamics.