A Lipschitz metric for the Hunter-Saxton equation
A Lipschitz metric for the Hunter-Saxton equation
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Hunter-Saxton 方程的 Lipschitz 度量
DOI：
10.1080/03605302.2018.1547744
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发表时间：
2019
影响因子：
1.9
通讯作者：
Antonio Carrillo J
中科院分区：
文献类型：
--
作者：
Antonio Carrillo J
分析了Hunter-Saxton（HS）方程直线上Cauchy问题保守解的稳定性.一般来说，HS方程的解在保持解本身的连续性的同时发展出具有陡峭梯度的奇点。为了获得唯一性，需要通过表示相关能量的测量来增加方程本身，并且解的分解与测量变得奇异的复杂相互作用相关联。在这篇文章中的主要结果是建设的Lipschitz度量，比较两个解决方案的HS方程与各自的初始数据。Lipschitz度量是基于Wasserstein度量的使用。
We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS 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 article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.