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

TL;DR

This paper establishes the analytical foundations for global weak solutions of the magnetic two-component Hunter-Saxton system, including explicit solution formulas and blow-up criteria, bridging geometric and PDE perspectives.

Contribution

It provides the first rigorous construction of global conservative weak solutions for the M2HS system from a PDE standpoint, extending the geometric framework.

Findings

Explicit solution formula in Lagrangian variables

Explicit blow-up time and criterion

Construction of global weak solutions

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}.