Year: 2026
Type: preprint
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
Source: arXiv (Cornell University)
Authors: Guelmame, Billel, Houamed, Haroune
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207713216