We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.

中文翻译：

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CAMASSA-HOLM 方程的 LIPSCHITZ 度量

我们分析了 Camassa-Holm (CH) 方程线上 Cauchy 问题的保守解的稳定性。通常，CH 方程的解会产生具有陡峭梯度的奇异性，同时保持解本身的连续性。为了获得唯一性，需要通过表示相关能量的度量来扩充方程本身，并且解决方案的分解与复杂的相互作用相关，其中度量变得奇异。本文的主要结果是构建了一个 Lipschitz 度量，该度量将 CH 方程的两个解与各自的初始数据进行了比较。Lipschitz 度量基于 Wasserstein 度量的使用。