Advances in Differential Equations

The Cauchy problem for the shallow water type equations in low regularity spaces on the circle

Yongsheng Li, Wei Yan, Xiaoping Zhai, and Yimin Zhang

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Abstract

In this paper, we investigate the Cauchy problem for the shallow water type equation \begin{align*} u_{t}+\partial_{x}^{3}u + \tfrac{1}{2}\partial_{x}(u^{2})+\partial_{x} (1-\partial_{x}^{2})^{-1}\left[u^{2}+\tfrac{1}{2} u_{x}^{2}\right]=0, \ \ x\in {\mathbf T}={\mathbf R}/2\pi \lambda, \end{align*} with low regularity data and $\lambda\geq1$. By applying the bilinear estimate in $W^{s}$, Himonas and Misiołek (Commun. Partial Diff. Eqns., 23 (1998), 123-139) proved that the problem is locally well-posed in $H^{s}([0, 2\pi))$ with $s\geq {1}/{2}$ for small initial data. In this paper, we show that, when $s < {1}/{2}$, the bilinear estimate in $W^{s}$ is invalid. We also demonstrate that the bilinear estimate in $Z^{s}$ is indeed valid for ${1}/{6} < s < {1}/{2}$. This enables us to prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$ with ${1}/{6} < s < {1}/{2}$ for small initial data.

Article information

Source
Adv. Differential Equations Volume 22, Number 5/6 (2017), 363-402.

Dates
First available in Project Euclid: 18 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ade/1489802455

Subjects
Primary: 35G25: Initial value problems for nonlinear higher-order equations

Citation

Yan, Wei; Li, Yongsheng; Zhai, Xiaoping; Zhang, Yimin. The Cauchy problem for the shallow water type equations in low regularity spaces on the circle. Adv. Differential Equations 22 (2017), no. 5/6, 363--402.https://projecteuclid.org/euclid.ade/1489802455


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