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

Levin Maier

Abstract

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.

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

Abstract

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 -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.

Paper Structure

This paper contains 24 sections, 27 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Magnetic systems and Mañé critical value
  3. Intermezzo: Magnetic systems
  4. Mañé's Critical Value of infinite-dimensional magnetic systems
  5. Magnetomorphism and totally magnetic submanifolds
  6. The magnetic system on SiL
  7. Toy model: the magnetic system on ${\mathbb{S}^3}$
  8. The magnetic system $({\mathbb{S}_{L^2}^{\infty}}, \mathcal{G}^{L^2}, {\mathrm{d}}\alpha)$
  9. Dynamical reduction from ${\mathbb{S}_{l^2}^{\infty}}$ to ${\mathbb{S}^3}$
  10. Mané's Critical Value and a Hopf--Rinow Theorem for $\left({\mathbb{S}_{L^2}^{\infty}}, \mathcal{G}^{L^2}, {\mathrm{d}}\alpha\right)$
  11. The Madelung transform as magnetomorphism
  12. The (M2HS) as a magnetic geodesic equation
  13. Statement of the Duality Theorem
  14. The Lorentz force
  15. Derivation of the magnetic geodesic equation
  16. ...and 9 more sections

Key Result

Theorem 2.11

ABM23 Let $(M,g_M,\sigma_M)$ be a magnetic system. Let $N\subset M$ be a closed, embedded submanifold. Denote by $\iota\colon N\to M$ the inclusion map and by $g_N:=\iota^*g_M$ and $\sigma_N:=\iota^*\sigma_M$ the pullback metric and magnetic field. The following statements are equivalent:

Figures (3)

  • Figure 1: This picture illustrates ABM23 and the fact that we can connect the Reeb orbit $\{z_2=0\}$ and the Reeb orbit $\{z_1=0\}$ with a unit speed magnetic geodesic of prescribed energy $k$ if and only if $k>\frac{1}{8}$, see Theorem \ref{['t:mane']}. (Picture by Ana Chavez Caliz.)
  • Figure 2: This is an illustration of \ref{['t: geometric interpret of blow ups']}. The main idea is that \ref{['(M2HS)']} exhibits a blowup if and only if the corresponding magnetic geodesic $\varPhi(\varphi, \tau)$ in $U$, shown as the green curve, intersects the boundary $\partial U$. (Illustration by Ana Chavez Caliz.)
  • Figure 3: Illustration of the proof of \ref{['t: global existence weak geodesic flow']}. The key idea is that, using \ref{['t: Madelungtransform as Magnetomorphims']} and \ref{['t: geometric interpret of blow ups']}, it suffices to extend the green curve $(\varPhi(\varphi), \varPhi(\tau))$ beyond the boundary. As $(\varPhi(\varphi), \varPhi(\tau))$ is, by \ref{['t: Madelungtransform as Magnetomorphims']} and ABM23, a magnetic geodesic in $(\mathcal{U}, \mathcal{G}^{L^2}, {\mathrm{d}}\alpha)$, and therefore also in $({\mathbb{S}_{L^2}^{\infty}}, \mathcal{G}^{L^2}, {\mathrm{d}}\alpha)$, the corresponding magnetic geodesic, represented by the blue curve, exists globally by \ref{['cor: dynamical reduction of SiL to Sd']}. This allows the extension of the green curve beyond $\partial \mathcal{U}$. (Illustration by Ana Chavez Caliz.)

Theorems & Definitions (66)

  • Definition 2.1
  • Remark 2.3
  • Definition 2.4: Mañé's critical value
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • Theorem 2.11
  • Proposition 3.1
  • Definition 3.3
  • ...and 56 more