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A Lipschitz metric for the Hunter-Saxton equation

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Title: A Lipschitz metric for the Hunter-Saxton equation
Authors: Carrillo de la Plata, J
Grunert, K
Holden, H
Item Type: 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.
Issue Date: 15-Feb-2019
Date of Acceptance: 28-Aug-2018
URI: http://hdl.handle.net/10044/1/63942
DOI: https://dx.doi.org/10.1080/03605302.2018.1547744
ISSN: 0360-5302
Publisher: Taylor & Francis
Journal / Book Title: Communications in Partial Differential Equations
Volume: 44
Issue: 4
Copyright 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 isproperly cited.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/P031587/1
Keywords: Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
Conservative solution
Hunter-Saxton equation
Lipschitz metric
Primary
Secondary
HYPERBOLIC VARIATIONAL EQUATION
GLOBAL CONSERVATIVE SOLUTIONS
LONG-TIME ASYMPTOTICS
EXISTENCE
UNIQUENESS
0101 Pure Mathematics
0102 Applied Mathematics
General Mathematics
Publication Status: Published
Online Publication Date: 2019-02-15
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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