scholarly journals A Lipschitz metric for the Hunter–Saxton equation

2019 ◽  
Vol 44 (4) ◽  
pp. 309-334 ◽  
Author(s):  
José Antonio Carrillo ◽  
Katrin Grunert ◽  
Helge Holden
Keyword(s):  
Lipschitz Metric

scholarly journals A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION

2020 ◽  
Vol 8 ◽  
Author(s):  
JOSÉ A. CARRILLO ◽  
KATRIN GRUNERT ◽  
HELGE HOLDEN
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Wasserstein Metric ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Lipschitz Metric

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.

scholarly journals Lipschitz metric for the periodic Camassa–Holm equation

2011 ◽  
Vol 250 (3) ◽  
pp. 1460-1492 ◽  
Author(s):  
Katrin Grunert ◽  
Helge Holden ◽  
Xavier Raynaud
Keyword(s):  
Holm Equation ◽  
Lipschitz Metric

scholarly journals Lipschitz metric for the Hunter–Saxton equation

2010 ◽  
Vol 94 (1) ◽  
pp. 68-92 ◽  
Author(s):  
Alberto Bressan ◽  
Helge Holden ◽  
Xavier Raynaud
Keyword(s):  
Lipschitz Metric
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