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Mathematics > Analysis of PDEs

arXiv:1805.02377 (math)
[Submitted on 7 May 2018 (v1), last revised 14 Aug 2018 (this version, v2)]

Title:Ill-posedness of the Camassa-Holm and related equations in the critical space

Authors:Zihua Guo, Xingxing Liu, Luc Molinet, Zhaoyang Yin
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Abstract:We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $p\in [1,\infty], r\in (1,\infty]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works (\cite{Danchin2,Byers,HHK}).
Comments: to appear JDE
Subjects: Analysis of PDEs (math.AP)
Cite as: arXiv:1805.02377 [math.AP]
  (or arXiv:1805.02377v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1805.02377
Related DOI: https://doi.org/10.1016/j.jde.2018.08.013

Submission history

From: Zihua Guo [view email]
[v1] Mon, 7 May 2018 07:19:27 UTC (10 KB)
[v2] Tue, 14 Aug 2018 04:54:38 UTC (11 KB)
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