# Well-posedness issues on the periodic modified Kawahara equation

### Chulkwang Kwak

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

## 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>21 $, and as a byproduct, we show the weak ill-posedness below $H_{21}(T)$, in the sense that the flow map fails to be uniformly continuous.

## Cite this article

Chulkwang Kwak, Well-posedness issues on the periodic modified Kawahara equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 2, pp. 373–416

DOI 10.1016/J.ANIHPC.2019.09.002