TY - JOUR

T1 - Well-posedness issues on the periodic modified Kawahara equation

AU - Kwak, Chulkwang

N1 - Funding Information:
C. K. is supported by FONDECYT Postdoctorado 2017 Proyecto No. 3170067 and project France-Chile ECOS-Sud C18E06.
Publisher Copyright:
© 2019 Elsevier Masson SAS

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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 L2(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 L2 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in Hs(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(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(T), s>[Formula presented], and as a byproduct, we show the weak ill-posedness below H[Formula presented](T), in the sense that the flow map fails to be uniformly continuous.

AB - 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 L2(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 L2 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in Hs(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(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(T), s>[Formula presented], and as a byproduct, we show the weak ill-posedness below H[Formula presented](T), in the sense that the flow map fails to be uniformly continuous.

KW - Global well-posedness

KW - Initial value problem

KW - Modified Kawahara equation

KW - Unconditional uniqueness

KW - Weak ill-posedness

UR - http://www.scopus.com/inward/record.url?scp=85073570598&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2019.09.002

DO - 10.1016/j.anihpc.2019.09.002

M3 - Article

AN - SCOPUS:85073570598

SN - 0294-1449

VL - 37

SP - 373

EP - 416

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 2

ER -