A Lipschitz metric for the Hunter–Saxton equation

Journal article

Abstract:: 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.

Files:: 1612.02961v1.pdf

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Publisher:: Taylor & Francis
Journal:: Communications in Partial Differential Equations More from this journal
Volume:: 44
Issue:: 4
Pages:: 309-334
Publication date:: 2019-02-15
Acceptance date:: 2018-08-29
DOI:: 10.1080/03605302.2018.1547744
EISSN:: 1532-4133
ISSN:: 0360-5302

Keywords:: Conservative solution

FFR

Hunter–Saxton equation

Lipschitz metric
Pubs id:: 1098230
Local pid:: pubs:1098230
Deposit date:: 2020-04-07

Copyright holder:: José Antonio Carrillo, Katrin Grunert and Helge Holden
Rights statement:: © 2019 The Author(s). Published by Taylor & Francis Group, LLC. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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