Strong Ill-Posedness in critical and subcritical regimes for the Hunter-Saxton equation

Abstract

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.