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May/June 2017 The Cauchy problem for the shallow water type equations in low regularity spaces on the circle
Yongsheng Li, Wei Yan, Xiaoping Zhai, Yimin Zhang
Adv. Differential Equations 22(5/6): 363-402 (May/June 2017).

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.

Citation

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Yongsheng Li. Wei Yan. Xiaoping Zhai. Yimin Zhang. "The Cauchy problem for the shallow water type equations in low regularity spaces on the circle." Adv. Differential Equations 22 (5/6) 363 - 402, May/June 2017.

Information

Published: May/June 2017
First available in Project Euclid: 18 March 2017

zbMATH: 1365.35016
MathSciNet: MR3625592

Subjects:
Primary: 35G25

Rights: Copyright © 2017 Khayyam Publishing, Inc.

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Vol.22 • No. 5/6 • May/June 2017
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