Methods and Applications of Analysis

Volume 23 (2016)

Number 2

Error estimate of the particle method for the $b$-equation

Pages: 119 – 154



Yong Duan (School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, China)

Jian-Guo Liu (Department of Physics and Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)


In this paper, we establish the optimal error estimate of the particle method for a family of nonlinear evolutionary partial differential equations, or the so-called $b$-equation. The $b$-equation, including the Camassa–Holm equation and the Degasperis–Procesi equation, has many applications in diverse scientific fields. The particle method is an approximation of the $b$-equation in Lagrangian representation. We also prove short-time existence, uniqueness and regularity of the Lagrangian representation of the $b$-equation.


Camassa–Holm equation, Degasperis–Procesi equation, Lagrangian representation, classical solution, particle method, peakon solutions, error estimate

2010 Mathematics Subject Classification

35B65, 35C08, 35D35, 65M15, 65M75

Full Text (PDF format)