Norm inflation for equations of KdV type with fractional dispersion

TL;DR

This paper proves that for certain fractional KdV equations, smooth small initial data can lead to solutions that rapidly become arbitrarily large in Sobolev norms, indicating ill-posedness.

Contribution

It establishes norm inflation phenomena for fractional dispersion KdV equations, extending known results to nonlocal and fractional settings.

Findings

Norm inflation occurs in fractional KdV equations.

Small initial data can lead to large solutions in short time.

Results hold in both periodic and non-periodic cases.

Abstract

We demonstrate norm inflation for nonlinear nonlocal equations, which extend the Korteweg-de Vries equation to permit fractional dispersion, in the periodic and non-periodic settings. That is, an initial datum is smooth and arbitrarily small in a Sobolev space, but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.