The Cauchy problem for the shallow water typ equations in low regularity   spaces on the circle
  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. 
Wei Yan, Yongsheng LI, Xiaoping Zhai, Yimin Zhang
math.AP