Sharp well-posedness results for the BBM equation

The regularized long-wave or BBM equationuₓ+uₗ+u uₗ-uₗ ₗ ₓ = 0was 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 fort>0. It is proven that this initial-value problem is globally wellposed in the L²-based Sobolev class Hˢ if s 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 C². Hence, it isconcluded that the BBM equation cannot be solved by iteration of abounded mapping leading to a fixed point in Hˢ-based spaces fors < 0. One is thus led to surmise that the initial-value problemfor the BBM equation is not even locally well posed in Hˢ fornegative values of s.