Sharp ill-posedness for the Hunter–Saxton equation on the line

The purpose of this paper is to give an exact division for the well-posedness and ill-posedness (non-existence) of the Hunter–Saxton equation on the line. Since the force term is not bounded in the classical Besov spaces (even in the classical Sobolev spaces), a new mixed space \(\mathcal {B}\) $\mathcal{B}$ is constructed to overcome this difficulty. More precisely, if the initial data \(u_0\in \mathcal {B}\cap \dot{H}^{1}(\mathbb {R}),\) $u_0\in \mathcal{B}\cap {\dot{H}}^1(\mathbb{R}),$ the local well-posedness of the Cauchy problem for the Hunter–Saxton equation is established in this space. Contrariwise, if \(u_0\in \mathcal {B}\) $u_0\in \mathcal{B}$ but \(u_0\notin \dot{H}^{1}(\mathbb {R}),\) $u_0\notin {\dot{H}}^1(\mathbb{R}),$ the norm inflation and hence the ill-posedness is presented. It’s worth noting that this norm inflation occurs in the low frequency part, which exactly leads to a non-existence result. Moreover, the above result clarifies a corollary with physical significance such that all the smooth solutions in \(L^{\infty }(0,T;L^{\infty }(\mathbb {R}))\) $L^{\infty }(0,T;L^{\infty }(\mathbb{R}))$ must have the \(\dot{H}^1\) ${\dot{H}}^1$ norm.