Geodesic
A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION
Forum of Mathematics, Sigma, Tome 8 (2020)

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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. 
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     author = {JOS\'E~A. CARRILLO and KATRIN GRUNERT and HELGE HOLDEN},
     title = {A {LIPSCHITZ} {METRIC} {FOR} {THE} {CAMASSA{\textendash}HOLM} {EQUATION}},
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JOSÉ A. CARRILLO; KATRIN GRUNERT; HELGE HOLDEN. A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.22

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