  The Kawahara and modified Kawahara equations are fifth-order KdV type
equations and have been derived to model many physical phenomena such as
gravity-capillary waves and magneto-sound propagation in plasmas. This paper
establishes the local well-posedness of the initial-value problem for Kawahara
equation in $H^s({\mathbf R})$ with $s>-\frac74$ and the local well-posedness
for the modified Kawahara equation in $H^s({\mathbf R})$ with $s\ge-\frac14$.
To prove these results, we derive a fundamental estimate on dyadic blocks for
the Kawahara equation through the $[k; Z]$ multiplier norm method of Tao
\cite{Tao2001} and use this to obtain new bilinear and trilinear estimates in
suitable Bourgain spaces.
