Well-posedness issues on the periodic modified Kawahara equation

Abstract

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 Hasimoto1970. We show in this paper some well-posedness results, mainly the global well-posedness in $L^2(\mathbb T)$. The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works 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 $L^2$