Hunter-Saxton方程在临界和次临界区域的强不适定性

中文摘要 AI 辅助

我们在实直线上研究Hunter-Saxton方程及其在能量空间$ L^\infty(\mathbb R)\cap \dot H^1(\mathbb R) $中的全局耗散解。我们证明在Lipschitz阈值及以下，通过Sobolev正则性的瞬时失效可得到强不适定性。更确切地说，对每个$s\in(1,\nicefrac32]$，我们构造$ u_0\in L^\infty(\mathbb R)\cap\dot H^1(\mathbb R)\cap\dot H^s(\mathbb R) $，其唯一的全局耗散解满足对每个$T>0$都有$u\notin C([0,T]; \dot H^s (\mathbb R))$。次临界和临界区域的构造存在显著差异：对$1<s<\nicefrac32$，我们叠加缩放后的、局域化的、具有越来越负斜率的泡；次临界缩放保持$\dot H^s$-可和性，而显式特征公式产生范数膨胀。在$s=\nicefrac32$处，缩放不产生小量，我们使用对数分布的紧支集多尺度剖面，其破裂时间收敛到零。单侧局域化原理和几乎正交性估计随后将单个剖面的膨胀传递到整个耗散解。

英文摘要

We study the Hunter--Saxton equation on the real line and its global dissipative solution in the energy space $ L^\infty(\mathbb R)\cap \dot H^1(\mathbb R). $ We prove strong ill-posedness through instantaneous failure of Sobolev regularity at and below the Lipschitz threshold. More precisely, for every $s\in(1,\nicefrac32]$, we construct $ u_0\in L^\infty(\mathbb R)\cap\dot H^1(\mathbb R)\cap\dot H^s(\mathbb R) $ whose unique global dissipative solution satisfies $u\notin C([0,T]; \dot H^s (\mathbb R))$, for every $T>0$. The constructions differ substantially in the subcritical and critical regimes. For $1<s<\nicefrac32$, we superpose rescaled, localized bubbles with increasingly negative slopes; subcritical scaling preserves $\dot H^s$-summability, while the explicit characteristic formula produces norm inflation. At $s=\nicefrac32$, where scaling yields no smallness, we use logarithmically distributed compactly supported multiscale profiles whose breaking times converge to zero. A one-sided localization principle and an almost-orthogonality estimate then transfer the inflation of individual profiles to the full dissipative solution.

发表机构

• New York University Abu Dhabi（阿布扎比纽约大学）
• Koç University（科奇大学）

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