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Well-posedness issues on the periodic modified Kawahara equation
- Title
- Well-posedness issues on the periodic modified Kawahara equation
- Authors
- Kwak, Chulkwang
- Ewha Authors
- 곽철광
- SCOPUS Author ID
- 곽철광
- Issue Date
- 2020
- Journal Title
- ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
- ISSN
- 0294-1449
1873-1430
- Citation
- ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE vol. 37, no. 2, pp. 373 - 416
- Keywords
- Modified Kawahara equation; Initial value problem; Global well-posedness; Unconditional uniqueness; Weak ill-posedness
- Publisher
- ELSEVIER
- Indexed
- SCIE; SCOPUS
- Document Type
- Article
- 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 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.
- DOI
- 10.1016/j.anihpc.2019.09.002
- Appears in Collections:
- 자연과학대학 > 수학전공 > Journal papers
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