关于两组分Hunter-Saxton系统的Mañé临界值及无限维磁Hopf-Rinow定理
On Mañé's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem
AI总结：
本文研究了两组分磁Hunter-Saxton系统的Mañé临界值及其无限维磁Hopf-Rinow定理，通过几何框架分析解的破裂行为并扩展解至全局保守弱解。
摘要：
在本文中，我们引入了一个非线性偏微分方程系统，即磁两组分Hunter-Saxton系统（M2HS）。该系统被定义为在无限维李群上配备右不变度量的磁测地方程，该度量是$\dot{H}^1$-度量，与无限维Fisher-Rao度量密切相关，且磁场由无限维接触型形式的导数构成。我们定义了在Hilbert流形上精确磁系统的Mañé临界值，并针对(M2HS)进行了显式计算。此外，我们建立了该磁系统的无限维Hopf-Rinow定理，其中Mañé临界值作为阈值，超过该阈值后Hopf-Rinow定理不再成立。这一几何框架使我们能够深入分析(M2HS)解的破裂行为。利用这一见解，我们通过引入并证明全局保守弱解的存在性来扩展解。此扩展通过将Madelung变换从等距变换扩展为磁变换，将磁系统嵌入到配备标准接触形式导数的无限维球面上的磁系统中得以实现。关键的是，此设置可通过动力学约简定理始终约简为一个完全磁的三维球面，从而更深入地理解底层动力学。
英文摘要：
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.
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