  We prove that the Kawahara equation is locally well-posed in $H^{-7/4}$ by
using the ideas of $\bar{F}^s$-type space \cite{GuoKdV}. Next we show it is
globally well-posed in $H^s$ for $s\geq -7/4$ by using the ideas of "I-method"
\cite{I-method}. Compared to the KdV equation, Kawahara equation has less
symmetries, such as no invariant scaling transform and not completely
integrable. The new ingredient is that we need to deal with some new
difficulties that are caused by the lack symmetries of this equation.
