This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on ), 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 . We show in this paper some well-posedness results, mainly the global well-posedness in . 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 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in , , due to the lack of -Strichartz estimate for arbitrary data, a slight modification, thus, is needed to attain the local well-posedness in . 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 , , and as a byproduct, we show the weak ill-posedness below , in the sense that the flow map fails to be uniformly continuous.
Cite this article
Chulkwang Kwak, Well-posedness issues on the periodic modified Kawahara equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 2, pp. 373–416DOI 10.1016/J.ANIHPC.2019.09.002