Well-posedness issues on the periodic modified Kawahara equation
- 주제(키워드) Modified Kawahara equation , Initial value problem , Global well-posedness , Unconditional uniqueness , Weak ill-posedness
- 주제(기타) Mathematics, Applied
- 설명문(일반) [Kwak, Chulkwang] Pontificia Univ Catolica Chile, Fac Matemat, Santiago, Chile; [Kwak, Chulkwang] Chonbuk Natl Univ, Inst Pure & Appl Math, Jeonju, South Korea
- 관리정보기술 faculty
- 등재 SCIE, SCOPUS
- 발행기관 ELSEVIER
- 발행년도 2020
- 총서유형 Journal
- URI http://www.dcollection.net/handler/ewha/000000172070
- 본문언어 영어
- Published As https://dx.doi.org/10.1016/j.anihpc.2019.09.002
초록/요약
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 L-2(T). The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works [69,60], 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 (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 (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 H-s (T), s > 1/2, and as a byproduct, we show the weak ill-posedness below H1/2 (T), in the sense that the flow map fails to be uniformly continuous. (C) 2019 Elsevier Masson SAS. All rights reserved.
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