In the present article, we prove the sharp local well-posedness and ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and ill-posed in $H^s(\mathbb{T})$ for $s-3/8$ given by Oh and Stefanov (2012) to the regularity threshold $H^{-1/2}(\mathbb{T})$. Similar ideas also establish the sharp local well-posedness in $H^{-1/2}(\mathbb{R})$ and ill-posedness below $H^{-1/2}$ for the nonperiodic case, which improves the result of Tsugawa and the author (2010) in $H^s(\mathbb{R})$ with $s>-1/2$ to the limiting regularity.

40 pages