2019/02/24 by Chulkwang Kwak, Kwak, Chulkwang · 1 citation
Earth and Planetary Sciences · Mathematics · #35G25 (Secondary) #35Q53 #76B15 (Primary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing
paper · pdf · doi:10.48550/arxiv.1902.08946
openalex publication_date 2019/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeHasimoto1970. We show in this paper some well-posedness results, mainly the global well-posedness in L2(\mathbb T). The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works \citeTT2004, 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 L2 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in Hs(\mathbb 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(\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 unconditional uniqueness in Hs(\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.