Norm-inflation results for the BBM equation

Considered here is the periodic initial-value problem for the regularized long-wave (BBM) equationut+ux+uux−uxxt=0.ut+ux+uux−uxxt=0. Adding to previous work in the literature, it is shown here that for any s<0s<0, there is smooth initial data that is small in the L2L2-based Sobolev spaces HsHs, but the solution emanating from it becomes arbitrarily large in arbitrarily small time. This so called norm inflation result has as a consequence the previously determined conclusion that this problem is ill-posed in these negative-norm spaces.