Communications in Mathematical Sciences

Volume 17 (2019)

Number 8

Regularity and singularity results for the dissipative Whitham equation and related surface wave equations

Pages: 2141 – 2190

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n8.a4

Authors

Qianyun Miao (School of Mathematical Sciences, Peking University, Beijing, China; and School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China)

Liutang Xue (Laboratory of Mathematics and Complex Systems (MOE), School ofMathematical Sciences, Beijing Normal University, Beijing, China)

Abstract

We consider the Cauchy problem for the Whitham equation and related surface wave equations with (fractional) dissipation. We prove global regularity results at the subcritical and critical dissipative cases by applying the method of modulus of continuity, and we show a finite-time singularity result at the supercritical dissipative case.

Keywords

Whitham equation, surface wave equation, global regularity, modulus of continuity, singularity

2010 Mathematics Subject Classification

35A01, 35Q35, 35Q86

Full Text (PDF format)

Received 17 October 2018

Accepted 12 July 2019

Published 3 February 2020