New Results on the Vibrating String with a Continuous Obstacle.

Aspects dynamiques du contact unilatéral entre un fluide et une structure

“…Pour en effectuer l'analyse, nous en considérons deux approximations régulières, dont les interprétations physiques des problèmes limites correspondent au problème de départ auquel on aurait ajouté la donnée d'une loi de restitution d'énergie. Ce phénomène rappelle fortement les résultats sur les modèles d'impacts de masses contre un obstacle obtenus par M. Schatzman [7] et L. Paoli et M. Schatzman [6] pour l'étude d'impacts ponctuels et A. Bamberger et M. Schatzman [1] pour une corde. C'est ce manque d'information relative à l'entropie du système, qui a constitué un point de départ à notre réflexion.…”

“…It is zero on the bottom Γ 0 . The (vertical) displacement of the body is denoted by q(t) = d(t)e z with d = d(t) ∈ R and it satisfies (1) where λ is the contact pressure of the body on the water and N the inward unit normal to the body. The displacement on the free surface η satisfies (2) and the velocity potential satisfies (5).…”

“…The non well-posedness of such problems is known. Even for finite dimensional models, these difficulties appear and we refer to the works of M. Schatzman and L. Paoli for contributions in the case of ODE [7,6], and to those of A. Bamberger and M. Schatzman [1] in the case of a vibrating string. In order to justify our model, let us introduce two approximate equations called respectively 'the weak model' and 'the strong model' that we present directly in terms of the first integral (in time) of the displacements.…”

Martingale solutions of a stochastic wave equation with reflection

“…Schatzman [11,12], and Bamberger and Schatzman [1] investigated vibration of a string which is constrained to stay on one side of an obstacle. With some condition on the geometry of the obstacle and the assumption of energy conservation, the existence and uniqueness of a solution to the Cauchy problem was established.…”

“…With some condition on the geometry of the obstacle and the assumption of energy conservation, the existence and uniqueness of a solution to the Cauchy problem was established. It is not known whether intricate geometric argument in [1,11,12] can be adapted to the equation with random noise. There are also results by other people on similar deterministic problems of hyperbolic nature.…”