Norm-inflation results for the BBM equation

TL;DR

This paper demonstrates that the periodic BBM equation exhibits norm inflation in negative Sobolev spaces, indicating ill-posedness for initial data with small norms in these spaces.

Contribution

It extends previous results by proving norm inflation for the BBM equation in all negative Sobolev spaces, confirming ill-posedness in these spaces.

Findings

Norm inflation occurs for all s<0 in Sobolev spaces.

Solutions can become arbitrarily large in arbitrarily small time.

The problem is ill-posed in negative Sobolev spaces.

Abstract

Considered here is the periodic initial-value probem for the regularized long-wave (BBM) equation \[u_t+u_x+uu_x-u_{xxt}=0.\] Adding to previous work in the literature, it is shown here that for any $s < 0$, there is smooth initial data that is small in the $L_2$-based Sobolev spaces $H^s$, but the solution emanating from it becomes arbitrarily large in arbitrarily small time. This so called {\it norm inflation} result has as a consequence the previously determined conclusion that this problem is ill-posed in these negative-norm spaces.