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Well-posedness for regularized nonlinear dispersive wave equations
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A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation
Sharp well-posedness 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 60607-7045 |
| 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 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 $s$.
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