Well-posedness of a higher order modified Camassa-Holm equation in spaces of low regularity. (English) Zbl 1239.35141

Li, Yongsheng; Yan, Wei; Yang, Xingyu

J. Evol. Equ. 10, No. 2, 465-486 (2010).

Summary: We consider the Cauchy problem for a higher-order modified Camassa-Holm equation. By using the Fourier restriction norm method introduced by J. Bourgain [Geom. Funct. Anal. 3, 209–262 (1993; Zbl 0787.35098)], we establish local well-posedness for initial data in \(H^s(\mathbb{R})\) with \(s>-n+5/4\), \(n\in {\mathbb{N}}^+\). As a consequence of the conservation of the energy \({\| u\|_{H^1(\mathbb{R})}}\), we have the global well-posedness for initial data in \({H^1(\mathbb{R})}\).

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

well-posedness; modified Camassa-Holm equation; Fourier restriction norm method; bilinear estimates

Citations:

Zbl 0787.35098

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