This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on $\mathbb{T}$), 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 $L^2(\mathbb{T})$. 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 $H^s(\mathbb{T})$, $s>0$, 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 $L^2(\mathbb{T})$. 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 $H^s(\mathbb{T})$, $s>\frac{1}{2}$, and as a byproduct, we show the weak ill-posedness below $H^{\frac{1}{2}}(\mathbb{T})$, in the sense that the flow map fails to be uniformly continuous.

本文关注修正的Kawahara方程的Cauchy问题 $\mathbb{Ť}$），这是在长波状态下平坦底部无限长的运河中毛细管重力波的模型[26]。我们发现在本文中的一些适定性结果，主要是全球适定性的${\mathrm{大号}}^2（\mathbb{Ť}）$。证明基本上依赖于高冈笃津的著作[60] [69]中引入的思想，该思想削弱了局部结果在立方相互作用中的非平凡共振（一种平滑效果）以及整体的适定性结果立即来自${\mathrm{大号}}^2$守恒定律。高冈津笃的想法的立即应用仅在$H^s（\mathbb{Ť}）$， $s>0$，由于缺乏 ${\mathrm{大号}}^4$-Strichartz估计为任意 ${\mathrm{大号}}^2$ 数据，因此，需要稍作修改才能在 ${\mathrm{大号}}^2（\mathbb{Ť}）$。据我们所知，这是周期性修正的Kwahara方程的第一个低正则性（全局）适定性结果。直接插值参数可确保条件中的无条件唯一性$H^s（\mathbb{Ť}）$， $s>\frac{\mathrm{1个}}{2}$，并且作为副产品，我们在下面显示了较弱的不适感 $H^{\frac{\mathrm{1个}}{2}}（\mathbb{Ť}）$从某种意义上说，流程图无法统一连续。