## Advances in Differential Equations

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

### 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

#### 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**

http://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. http://projecteuclid.org/euclid.ade/1489802455.