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In this paper, we investigate the Cauchy problem for the shallow water typeequation \[ u_{t}+\partial_{x}^{3}u  + \frac{1}{2}\partial_{x}(u^{2})+\partial_{x}  (1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0,x\in{\mathbf T}=\R/2\pi  \lambda \] 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 thispaper improves the result of case $j=1$ of Himonas and Misiolek (Communicationsin Partial Differential Equations, 23(1998), 123-139.). The new ingredients aresome new function spaces and some new Strichartz estimates.

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