Well-posedness issues on the periodic modified Kawahara equation

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.