## 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 [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>[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.

Original language | English |
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Pages (from-to) | 373-416 |

Number of pages | 44 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 37 |

Issue number | 2 |

DOIs | |

State | Published - 1 Mar 2020 |

### Bibliographical note

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

## Keywords

- Global well-posedness
- Initial value problem
- Modified Kawahara equation
- Unconditional uniqueness
- Weak ill-posedness