Sharp well-posedness results for the BBM equation

Abstract

The regularized long-wave or BBM equation$u_{t}+u_{x}+u u_{x}-u_{x x t} = 0$was derived as a model for the unidirectional propagation oflong-crested, surface water waves. It arises in other contexts aswell, and is generally understood as an alternative to theKorteweg-de Vries equation. Considered here is the initial-valueproblem wherein $u$ is specified everywhere at a given time $t = 0$,say, and inquiry is then made into its further development for$t>0$. It is proven that this initial-value problem is globally wellposed in the $L^2$-based Sobolev class $H^s$ if $s \geq 0$.Moreover, the map that associates the relevant solution to giveninitial data is shown to be smooth. On the other hand, if $s < 0$,it is demonstrated that the correspondence between initial data andputative solutions cannot be even of class $C^2$. Hence, it isconcluded that the BBM equation cannot be solved by iteration of abounded mapping leading to a fixed point in $H^s$-based spaces for$s < 0$. One is thus led to surmise that the initial-value problemfor the BBM equation is not even locally well posed in $H^s$ fornegative values of $s$.