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August 2018, 23(6): 2545-2592. doi: 10.3934/dcdsb.2018067
The modified Camassa-Holm equation in Lagrangian coordinates
Yu Gao 1,2,, and Jian-Guo Liu 2,
Department of Mathematics of Harbin Institute of Technology, Harbin, 150001, China
Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA
* Corresponding author: Yu Gao
Received June 2017 Revised September 2017 Published February 2018
Fund Project: The second author is supported by KI-Net NSF RNMS (Grant No. 1107444) and NSF DMS (Grant No. 1514826)
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: $T_{max}≥ \frac{1}{||m_0||_{L^∞}||m_0||_{L^1}}.$ And there is a unique solution $X(ξ, t)$ to the Lagrange dynamics which is a strictly monotonic function of $ξ$ for any $t∈[0, T_{max})$: $X_ξ(·, t)>0$. As $t$ approaching $T_{max}$, we prove that the classical solution $m(·, t)$ in Eulerian coordinates has a unique limit $m(·, T_{max})$ in Radon measure space and there is a point $ξ_0$ such that $X_ξ(ξ_0, T_{max}) = 0$ which means $T_{max}$ is an onset time of collisions 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.
Keywords: Integrable system, Lagrange dynamics, finite time blow up, local classical solutions, global weak solutions.
Mathematics Subject Classification: Primary: 35G25, 35B44; Secondary: 35D30.
Citation: Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
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Figure 1. At $T_{max}$, $X_\xi(\cdot,T_{max})\geq0$ and $X_\xi(\xi,T_{max}) = 0$ for $\xi\in\{\xi_1,\xi_4\}\cup[\xi_{21},\xi_{22}]\cup[\xi_{31},\xi_{32}]$. $F_{T_{max}} = \{x_1,x_2,x_3,x_4\}$ and $\widehat{F}_{T_{max}} = \{x_2,x_3\}$
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