Norm inflation and ill-posedness for the Fornberg-Whitham equation

TL;DR

This paper demonstrates that the Cauchy problem for the Fornberg-Whitham equation exhibits ill-posedness through norm inflation in certain Besov spaces, indicating instability for specific initial data.

Contribution

It establishes the ill-posedness of the Fornberg-Whitham equation in particular function spaces by proving norm inflation phenomena.

Findings

Norm inflation occurs in specified Besov spaces.

The Cauchy problem is not locally well-posed in these spaces.

Ill-posedness is demonstrated for special initial data.

Abstract

In this paper, we prove that the Cauchy problem for the Fornberg-Whitham equation is not locally well-posed in $B^s_{p,r}(\R)$ with $(s,p,r)\in (1,1+\frac1p)\times[2,\infty)\times [1,\infty]$ or $(s,p,r)\in \{1\}\times[2,\infty)\times [1,2]$ by showing norm inflation phenomena of the solution for some special initial data.