American Institute of Mathematical Sciences

$ u_{t}+u_{x}+u u_{x}-u_{x x t} = 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 $L^2$-based Sobolev class $H^s$ if $s \geq 0$. 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 $C^2$. Hence, it is concluded that the BBM equation cannot be solved by iteration of a bounded mapping leading to a fixed point in $H^s$-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 $H^s$ for negative values of $s$.

$ u_t+u_x+g(u)_x+Lu_t=0, \qquad x\in\mathbb R,\quad t>0,$
$u(x,0)=u_0(x), \qquad x\in\mathbb R, (0.1)$

where $u=u(x,t)$ is a real-valued function, $L$ is a Fourier multiplier operator with real symbol $\alpha(\xi),$ say, and $g$ is a smooth, real-valued function of a real variable. Equations of this form arise as models of wave propagation in a variety of physical contexts. Here, fundamental issues of local and global well-posedness are established for $L_p$, $H^s$ and bore-like or kink-like initial data. In the special case where $\alpha(\xi)=|\xi|^{r}$ wherein $r>1$ and $g(u)=1/2u^2,$ (0.1) is globally well-posed in time if $s$ and $r$ satisfy a simple algebraic relation.