  This paper is concerned with the Cauchy problem of the modified Kawahara
equation (posed on $\mathbb T$), which is well-known as a model of
capillary-gravity waves in an infinitely long canal over a flat bottom in a
long wave regime \cite{Hasimoto1970}. We show in this paper some well-posedness
results, mainly the \emph{global well-posedness} in $L^2(\mathbb T)$. The proof
basically relies on the idea introduced in Takaoka-Tsutsumi's works
\cite{TT2004, NTT2010}, which weakens the non-trivial resonance in the cubic
interactions (a kind of smoothing effect) for the local result, and the global
well-posedness result immediately follows from $L^2$ conservation law. An
immediate application of Takaoka-Tsutsumi's idea is available only in
$H^s(\mathbb T)$, $s > 0$, due to the lack of $L^4$-Strichartz estimate for
arbitrary $L^2$ data, a slight modification, thus, is needed to attain the
local well-posedness in $L^2(\mathbb T)$. This is the first low regularity
(global) well-posedness result for the periodic modified Kwahara equation, as
far as we know. A direct interpolation argument ensures the \emph{unconditional
uniqueness} in $H^s(\mathbb T)$, $s > \frac12$, and as a byproduct, we show the
weak ill-posedness below $H^{\frac12}(\mathbb T)$, in the sense that the flow
map fails to be uniformly continuous.
