Sharp Well-posedness Results for the BBM Equation

The regularized long-wave or BBM equation ut + ux + uux − uxxt = 0 was derived as a model for the unidirectional propagation of long-crested, surface water waves. It arises in other contexts as well, and is generally understood as an alternative to the Korteweg-de Vries equation. Considered here is the initial-value problem 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 well posed in the L-based Sobolev class H if s ≥ 0. Moreover, the map that associates the relevant solution to given initial data is shown to be smooth. On the other hand, if s < 0, it is demonstrated that the correspondence between initial data and putative solutions cannot be even of class C. Hence, it is concluded that the BBM equation cannot be solved by iteration of a bounded mapping leading to a fixed point in H-based spaces for s < 0. One is thus led to surmise that the initial-value problem for the BBM equation is not even locally well posed in H for negative values of s.