2017/05/18 by Yu Gao, Gao, Yu, Jian‐Guo Liu +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1705.06562
openalex publication_date 2017/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data m0, we show that classical solutions to this equation blow up in finite time Tmax. Before Tmax, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: Tmax≥ \frac1||m0||L^∞||m0||L1. 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,Tmax): Xξ(⋅,t)>0. As t approaching Tmax, we prove that classical solution m(⋅ ,t) in Eulerian coordinate has a unique limit m(⋅,Tmax) in Radon measure space and there is a point ξ0 such that Xξ(ξ0,Tmax)=0 which means Tmax is an onset time of collision of characteristics. We also show that in some cases peakons are formed at Tmax. After Tmax, we regularize the Lagrange dynamics to prove global existence of weak solutions m in Radon measure space.