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# Methods and Applications of Analysis

## Volume 23 (2016)

### Number 2

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

Pages: 119 – 154

DOI: http://dx.doi.org/10.4310/MAA.2016.v23.n2.a1

#### Authors

#### Abstract

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.

#### Keywords

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

Published 30 June 2016