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 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 [26]. We show in this paper some well-posedness results, mainly the global well-posedness in L2(T). The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works [60,69], 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 L2 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in Hs(T), s>0, due to the lack of L4-Strichartz estimate for arbitrary L2 data, a slight modification, thus, is needed to attain the local well-posedness in L2(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 unconditional uniqueness in Hs(T), s>12, and as a byproduct, we show the weak ill-posedness below H12(T), in the sense that the flow map fails to be uniformly continuous.