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 $${\dot{H}}^1$$-...
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
06 June 2026
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| In: |
Communications in mathematical physics
Year: 2026, Volume: 407, Issue: 7, Pages: 1-34 |
| ISSN: | 1432-0916 |
| DOI: | 10.1007/s00220-026-05651-z |
| Online Access: | Verlag, kostenfrei, Volltext: https://doi.org/10.1007/s00220-026-05651-z
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| Author Notes: | L. Maier |
| Summary: | 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 $${\dot{H}}^1$$-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. |
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| Item Description: | Veröffentlicht: 06. Juni 2026 Gesehen am 21.08.2026 |
| Physical Description: | Online Resource |
| ISSN: | 1432-0916 |
| DOI: | 10.1007/s00220-026-05651-z |