arXiv 1902.089462019-02math.AP

周期修正Kawahara方程的适定性问题

Well-posedness issues on the periodic modified Kawahara equation

Chulkwang Kwak

备注：: 39 pages, revised the proof of the local well-posedness in $L^2$, accepted for publication in Annales de l'Institut Henri Poincare / Analyse non lineaire

摘要：

本文研究了修正的Kawahara方程的Cauchy问题（提出在$\mathbb T$上），该方程是著名的无限长渠道中的重力毛细波模型，在长波区为\cite{Hasimoto1970}。本文给出了一些适定性结果，主要是$L^2(\mathbb T)$中的\emph{global well-posedness}.证明基本上依赖于Takaoka-Tsutsumi的著作\cite{TT2004, NTT2010}中引入的思想，该思想削弱了局部结果中立方相互作用中的非平凡共振（一种平滑效应），而全局适定性结果直接由$L^2$守恒律得出. Takaoka-Tsutsumi思想的直接应用仅在$H^s(\mathbb T)$，$s > 0$中可用，由于缺乏对任意$L^2$数据的$L^4$-Eschhartz估计，因此，在$L^2(\mathbb T)$中需要稍微修改以获得局部适定性.这是第一个低正则性（全球）适定性结果的周期修改Kwahara方程，据我们所知。一个直接的插值参数，确保\emph{unconditional uniqueness}在$H^s(\mathbb T)$，$s > \frac12$，作为一个副产品，我们显示弱不适定性低于$H^{\frac12}(\mathbb T)$，在这个意义上说，流图未能一致连续。

英文摘要：

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 \cite{Hasimoto1970}. We show in this paper some well-posedness results, mainly the \emph{global well-posedness} in $L^2(\mathbb T)$. The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works \cite{TT2004, NTT2010}, 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 \emph{unconditional uniqueness} in $H^s(\mathbb T)$, $s > \frac12$, and as a byproduct, we show the weak ill-posedness below $H^{\frac12}(\mathbb T)$, in the sense that the flow map fails to be uniformly continuous.

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