# The modified Camassa-Holm equation in Lagrangian coordinates

## TL;DR

This paper investigates the finite-time blow-up and solution behavior of the modified Camassa-Holm equation in Lagrangian coordinates, establishing lifespan, collision phenomena, and the formation of peakons.

## Contribution

It provides a detailed analysis of solution lifespan, blow-up, and peakon formation for the modified Camassa-Holm equation in Lagrangian form, including regularization for global weak solutions.

## Key findings

- Classical solutions blow up in finite time for certain initial data.
- Lifespan of solutions is bounded below by inverse of initial data norms.
- Peakons can form at the blow-up time.

## Abstract

In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data $m_0$, we show that classical solutions to this equation blow up in finite time $T_{max}$. Before $T_{max}$, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: $T_{max}\geq \frac{1}{||m_0||_{L^\infty}||m_0||_{L^1}}.$ And there is a unique solution $X(\xi,t)$ to the Lagrange dynamics which is a strictly monotonic function of $\xi$ for any $t\in[0,T_{max})$: $X_\xi(\cdot,t)>0$. As $t$ approaching $T_{max}$, we prove that classical solution $m(\cdot ,t)$ in Eulerian coordinate has a unique limit $m(\cdot,T_{max})$ in Radon measure space and there is a point $\xi_0$ such that $X_\xi(\xi_0,T_{max})=0$ which means $T_{max}$ is an onset time of collision of characteristics. We also show that in some cases peakons are formed at $T_{max}$. After $T_{max}$, we regularize the Lagrange dynamics to prove global existence of weak solutions $m$ in Radon measure space.