The Cauchy problem for a higher order shallow water type equation on the circle

TL;DR

This paper establishes local and global well-posedness results for a higher order shallow water type equation on the circle, extending previous work and using the I-method for global results.

Contribution

It proves local well-posedness for initial data in low regularity Sobolev spaces and extends this to global well-posedness using the I-method, improving prior results.

Findings

Local well-posedness in H^s for s ≥ -(j-2)/2

Global well-posedness in H^s for (2j+1 - j^2)/(2j+1) < s ≤ 1

Extension of previous results on shallow water equations

Abstract

In this paper, we investigate the Cauchy problem for a higher order shallow water type equation \begin{eqnarray*} u_{t}-u_{txx}+\partial_{x}^{2j+1}u-\partial_{x}^{2j+3}u+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}=0, \end{eqnarray*} where $x\in \mathbf{T}=\mathbf{R}/2\pi$ and $j\in N^{+}.$ Firstly, we prove that the Cauchy problem for the shallow water type equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -\frac{j-2}{2}$ for arbitrary initial data. By using the $I$-method, we prove that the Cauchy problem for the shallow water type equation is globally well-posed in $H^{s}(\mathbf{T})$ with $\frac{2j+1-j^{2}}{2j+1}<s\leq 1.$ Our results improve the result of A. A. Himonas, G. Misiolek (Communications in partial Differential Equations, 23(1998), 123-139;Journal of Differential Equations, 161(2000), 479-495.)