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

TL;DR

This paper demonstrates that the Cauchy problem for the rod equation exhibits norm inflation and ill-posedness in certain Sobolev spaces, using explicit smooth initial data to show solutions can become arbitrarily large quickly.

Contribution

It introduces a new method using explicit smooth initial data to prove ill-posedness via norm inflation for the rod equation in low-regularity Sobolev spaces.

Findings

The problem is ill-posed in $H^s(\R)$ for $1< s<3/2$.

Solutions can become arbitrarily large in short time despite small initial data.

A new approach to proving ill-posedness via explicit initial data is developed.

Abstract

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 $H^s(\R)$ with $1< s<3/2$ in the sense of {\it norm inflation}, i.e., an initial data is smooth and arbitrarily small in $H^s(\R)$ with $1< s<3/2$, but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.