Differential and Integral Equations

Nonuniform dependence and well posedness for the periodic Hunter-Saxton equation

Curtis Holliman

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

It is proved that the flow map for the Hunter-Saxton (HS) equation from the homogeneous Sobolev space $\dot{H}^s({\mathbb{T}})$ into the space $C([0,T], \dot{H}^s({\mathbb{T}}))$ is continuous but not uniformly continuous on bounded subsets. To demonstrate this sharpness of continuity, two sequences of bounded solutions to the HS equation are constructed whose distance at the initial time converges to zero and whose distance at any later time is bounded from below by a positive constant. To achieve this result, appropriate approximate solutions are chosen and then the actual solutions are found by solving the Cauchy problem with initial data taken to be the value of approximate solutions at time zero. Then, using well-posedness estimates, it is shown that the difference between solutions and approximate solutions is negligible.

Article information

Source
Differential Integral Equations, Volume 23, Number 11/12 (2010), 1159-1194.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019079

Mathematical Reviews number (MathSciNet)
MR2742484

Zentralblatt MATH identifier
1240.35454

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Citation

Holliman, Curtis. Nonuniform dependence and well posedness for the periodic Hunter-Saxton equation. Differential Integral Equations 23 (2010), no. 11/12, 1159--1194. https://projecteuclid.org/euclid.die/1356019079


Export citation