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

TL;DR

This paper establishes local well-posedness for a shallow water type equation on the circle with low regularity initial data, introducing new function spaces and Strichartz estimates to improve previous results.

Contribution

It proves local well-posedness in low regularity spaces for the shallow water equation, extending prior work by employing novel function spaces and estimates.

Findings

Well-posedness for s in (1/6, 1/2) with small initial data

Invalidity of bilinear estimates in X_{s,b} for s<1/2

Introduction of new function spaces and Strichartz estimates

Abstract

In this paper, we investigate the Cauchy problem for the shallow water type equation \[ u_{t}+\partial_{x}^{3}u + \frac{1}{2}\partial_{x}(u^{2})+\partial_{x} (1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0,x\in {\mathbf T}=\R/2\pi \lambda \] with low regularity data in the periodic settings and $\lambda\geq1$. We prove that the bilinear estimate in $X_{s,b}$ with $s<\frac{1}{2}$ is invalid. We also prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$ with $\frac{1}{6}<s<\frac{1}{2}$ for small initial data. The result of this paper improves the result of case $j=1$ of Himonas and Misiolek (Communications in Partial Differential Equations, 23(1998), 123-139.). The new ingredients are some new function spaces and some new Strichartz estimates.