Methods and Applications of Analysis
Volume 23 (2016)
Error estimate of the particle method for the $b$-equation
Pages: 119 – 154
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