Norm inflation and low-regularity ill-posedness for the rod equation

In this paper, we consider the Cauchy problem for the rod equation in the line. By constructing an explicit smooth initial data, we present a new method to prove that this problem is ill-posed in $H^s(\mathbb{R})$ with $1<s<3\mathrm{/}2$ in the sense of {\it norm inflation}, i.e., an initial data is smooth and arbitrarily small in $H^s(\mathbb{R})$ with $1<s<3\mathrm{/}2$, but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.