Sharp well-posedness results for the BBM equation
Jerry Bona Nikolay Tzvetkov
The regularized long-wave or BBM equation

$ 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 $s$.

keywords: BBM-equation global well-posedness sharp well-posedness
On the transverse instability of one dimensional capillary-gravity waves
Frédéric Rousset Nikolay Tzvetkov
This text represents the content of a talk given by the second author at ENS Paris on January 28, 2009 at the conference "Mathematics and Oceanography". We are grateful to David Gérard-Varet, David Lannes and Laure Saint-Raymond for the kind invitation to present our recent work on the water waves. The proofs of the results announced here will appear in [18].
keywords: water-waves. free boundary problems solitary waves stability

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