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A twophase problem for capillarygravity waves and the BenjaminOno equation
Sharp wellposedness results for the BBM equation
1.  Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago , 851 S. Morgan Street MC 249, Chicago, Illinois 606077045 
2.  Laboratoire Paul Painlevé, Bâtiment M2, Cité scientifique, 59655 Villeneuve D'ascq Cedex, France 
$ u_{t}+u_{x}+u u_{x}u_{x x t} = 0 $
was derived as a model for the unidirectional propagation of longcrested, surface water waves. It arises in other contexts as well, and is generally understood as an alternative to the Kortewegde Vries equation. Considered here is the initialvalue 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 initialvalue 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 initialvalue problem for the BBM equation is not even locally well posed in $H^s$ for negative values of $s$.
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