A new Green-Naghdi model in the Camassa-Holm regime and full justification of asymptotic models for the propagation of internal waves

Vincent Duchêne (1) , Samer Israwi (2, 3) , Raafat Talhouk (4)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Faculté des Sciences
3 CRAMS - Center for Research in Applied Mathematics and Statistics
4 Laboratoire de mathématiques

This study deals with asymptotic models for the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. We present a new Green-Naghdi type model in the Camassa-Holm (or medium amplitude) regime. This model is fully justified, in the sense that it is consistent, well-posed, and that its solutions remain close to exact solutions of the full Euler system with corresponding initial data. Moreover, our system allows to fully justify any well-posed and consistent lower order model; and in particular the so-called Constantin-Lannes approximation, which extends the classical Korteweg-de Vries equation in the Camassa-Holm regime.

Mots clés: asymptotic models; internal gravity waves; shallow water

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. A new Green-Naghdi model in the Camassa-Holm regime and full justification of asymptotic models for the propagation of internal waves. 2013. ⟨hal-00814008v1⟩