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On Global Weak Solutions for the Magnetic Two-Component Hunter-Saxton System

Levin Maier

TL;DR

This work develops a rigorous PDE framework for the magnetic two-component Hunter–Saxton system (M2HS), connecting its geometric origin as a magnetic geodesic flow to a concrete analytical theory. It first derives an explicit solution formula in Lagrangian variables via a Riccati reduction, yielding a sharp blow-up criterion and an explicit blow-up time, and then establishes a global theory of weak solutions by introducing a relaxed configuration space $\mathcal{M}_{\dot{H}^1}$ and a corresponding weak magnetic geodesic flow. The authors prove global existence of unit-speed weak magnetic geodesics on $\mathcal{M}_{\dot{H}^1}$ and transfer these to Eulerian variables to obtain global conservative weak solutions of (M2HS) with almost-everywhere conservation of the $\dot{H}^1$ energy and the contact angle. In the large-magnetic-field limit ($s\to\infty$), blow-up is ruled out for fixed initial data, highlighting a regime where the weak flow remains globally regular; the results extend known weak theories for $s=0$ and $\rho\equiv s$ to the fully coupled, magnetically influenced system.

Abstract

We study the magnetic two-component Hunter-Saxton system (M2HS), which was recently derived in \cite{M24} as a magnetic geodesic equation on an infinite-dimensional configuration space. While the geometric framework and the global weak flow were outlined there, the present paper provides the analytical foundations of this construction from the PDE perspective. First, we derive an explicit solution formula in Lagrangian variables via a Riccati reduction, yielding an alternative proof of the blow-up criterion together with an explicit expression for the blow-up time. Second, we rigorously construct global conservative weak solutions by developing the analytic theory of the relaxed configuration space and the associated weak magnetic geodesic flow, thereby realizing the geometric program proposed in \cite{M24}.

On Global Weak Solutions for the Magnetic Two-Component Hunter-Saxton System

TL;DR

This work develops a rigorous PDE framework for the magnetic two-component Hunter–Saxton system (M2HS), connecting its geometric origin as a magnetic geodesic flow to a concrete analytical theory. It first derives an explicit solution formula in Lagrangian variables via a Riccati reduction, yielding a sharp blow-up criterion and an explicit blow-up time, and then establishes a global theory of weak solutions by introducing a relaxed configuration space and a corresponding weak magnetic geodesic flow. The authors prove global existence of unit-speed weak magnetic geodesics on and transfer these to Eulerian variables to obtain global conservative weak solutions of (M2HS) with almost-everywhere conservation of the energy and the contact angle. In the large-magnetic-field limit (), blow-up is ruled out for fixed initial data, highlighting a regime where the weak flow remains globally regular; the results extend known weak theories for and to the fully coupled, magnetically influenced system.

Abstract

We study the magnetic two-component Hunter-Saxton system (M2HS), which was recently derived in \cite{M24} as a magnetic geodesic equation on an infinite-dimensional configuration space. While the geometric framework and the global weak flow were outlined there, the present paper provides the analytical foundations of this construction from the PDE perspective. First, we derive an explicit solution formula in Lagrangian variables via a Riccati reduction, yielding an alternative proof of the blow-up criterion together with an explicit expression for the blow-up time. Second, we rigorously construct global conservative weak solutions by developing the analytic theory of the relaxed configuration space and the associated weak magnetic geodesic flow, thereby realizing the geometric program proposed in \cite{M24}.
Paper Structure (20 sections, 13 theorems, 98 equations)

This paper contains 20 sections, 13 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The equations and motivation
  3. Main contributions of this work
  4. Outline of the article
  5. Acknowledgments:
  6. Preliminaries: geometric origin of (M2HS)
  7. Background: Magnetic geodesics
  8. \ref{['eq:M2HS']} as magnetic geodesic equation
  9. Explicit solution formula for the magnetic two-component Hunter--Saxton system
  10. The (M2HS) system in Lagrangian variables
  11. Reduction to a Riccati equation
  12. Explicit representation formula and blow-up criterion
  13. Asymptotic absence of blow-up for strong magnetic field
  14. Global conservative weak solutions of the magnetic two-component Hunter--Saxton system
  15. Preparation: the closure of $G^m$ and its tangent structure
  16. ...and 5 more sections

Key Result

Theorem 2.1

A $C^2$-curve $(\varphi,\tau): [0,T)\longrightarrow G^m$, where $T>0$ is the maximal existence time, is a magnetic geodesic of the system $\left(G^{m}, \mathcal{G}^{\dot{H}^1}, \varPhi^{*}\mathrm{d}\alpha\right)$ with $m > \frac{5}{2}$ if and only if is a solution of the magnetic two-component Hunter--Saxton system eq:M2HS, that is: where $c^2$, the $\dot{H}^1$-energy of the system, is a conserv

Theorems & Definitions (34)

  • Theorem 2.1: M24
  • Proposition 1
  • Lemma 1
  • Theorem 3.1: Explicit solution formula and blow-up time
  • Remark 1
  • Remark 2
  • proof : Proof of Theorem \ref{['thm:explicit_formula']}
  • Lemma 2
  • proof
  • Definition 1: Relaxed configuration space
  • ...and 24 more