Well-posedness issues on the periodic modified Kawahara equation

Kwak, Chulkwang

doi:10.1016/j.anihpc.2019.09.002

Kwak, Chulkwang ^1,²

¹ Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile
² Institute of Pure and Applied Mathematics, Chonbuk National University, Republic of Korea

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 373-416.

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Résumé

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 [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 $L^{2}$ 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 $H^{s} (T)$ , $s > \frac{1}{2}$ , and as a byproduct, we show the weak ill-posedness below $H^{\frac{1}{2}} (T)$ , in the sense that the flow map fails to be uniformly continuous.

DOI : 10.1016/j.anihpc.2019.09.002

Classification : 35Q53, 76B15, 35G25
Mots-clés : Modified Kawahara equation, Initial value problem, Global well-posedness, Unconditional uniqueness, Weak ill-posedness

Affiliations des auteurs :

Kwak, Chulkwang ^{1, 2}

¹ Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile
² Institute of Pure and Applied Mathematics, Chonbuk National University, Republic of Korea

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     title = {Well-posedness issues on the periodic modified {Kawahara} equation},
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Kwak, Chulkwang. Well-posedness issues on the periodic modified Kawahara equation. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 373-416. doi : 10.1016/j.anihpc.2019.09.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.09.002/

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