On the Cauchy problem of the modified Hunter-Saxton equation

Yongsheng  Mi; Chunlai Mu; Pan  Zheng

doi:10.3934/dcdss.2016084

Article Contents

2016, Volume 9, Issue 6: 2047-2072. Doi: 10.3934/dcdss.2016084

This issue Previous Article Second-order slip flow of a generalized Oldroyd-B fluid through porous medium Next Article Scattering theory for energy-supercritical Klein-Gordon equation

On the Cauchy problem of the modified Hunter-Saxton equation

1.
College of Mathematics and Computer Sciences, Yangtze Normal University, Fuling 408100, Chongqing
2.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331
3.
Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065

Received: June 30, 2015

Revised: August 31, 2016

Published: November 2016

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Abstract

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation, which was proposed by by J. Hunter and R. Saxton [SIAM J. Appl. Math. 51(1991) 1498-1521]. Using the approximate solution method, the local well-posedness of the model equation is obtained in Sobolev spaces $H^{s}$ with $s > 3/2$, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. However, if a weaker $H^{r}$-topology is used then it is shown that the solution map becomes Hölder continuous in $H^{s}$.

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Mathematics Subject Classification: 35B30, 35Q53.

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