An initial - boundary - value problem for the Korteweg - deVries equation posed on a nite

R esum e. La propagation unidirectionnelle d'ondes de faible amplitude et de grande longueur d'onde est d ecrite, dans de nombreux syst emes physiques, par l' equation de Korteweg-de Vries. L'objet de ce travail est de proposer un probl eme mixte bien pos e lorsque le do-maine spatial est born e. Plus pr ecis ement nous etablissons l'existence de solutions locales en temps pour des donn ees initiales dans l'espace de Sobolev H 1 ainsi que l'existence de solutions globales pour donn ees initiales petites dans cet espace. Dans un second temps, nous mettons en evidence des eeets r egularisants globaux forts. Abstract. The Korteweg-de Vries equation occurs as a model for unidirectional propagation of small amplitude long waves in numerous physical systems. The aim of this work is to propose a well posed mixed initial-boundary-values problem when the spacial domain is of nite extent. More precisely, we establish local existence of solutions for arbitrary initial data in the Sobolev space H 1 and global existence for small initial data in this space. In a second step we show global strong regularizing eeects. 1 Setting of the problem and main results In 1895 Korteweg and de Vries 18] have introduced the equation that bears their names in the context of uni-directional water vaves propagating in an innnite chanel. These authors have considered a layer of incompressible uid over a at bottom and assumed that the ow was irrotational (see Colin-Dias and Ghidaglia 9] and 10] for rotational ows). Under the innuence of gravity, the motion of surface waves leads to a system of p.d.e.'s in the variables (x; t) and v(x; t) where is the wave height measured from an undisturbed water level and v(x; t) is the spatial derivative of the restriction to the surface of the velocity potential. Considering waves with wavelength l and amplitude a which satisfy a = O(") and 1 l = O(" ?1=2) (where " denotes a small parameter) and in the hypothesis of uni-directional waves, one gets, at the leading order in " and after scaling transforms the following equation for : t + hh x + "h 3 12 xxx + 3 2 x = 0: (1.1) Here h denotes the ((nite) depth of the uid. In the case of an innnite chanel (i.e. x 2 IR) it is possible to perform further scaling and change in the independent variables x and …