Well-posedness of the Green-Naghdi and Boussinesq-Peregrine systems

Vincent Duchêne (1) , Samer Israwi (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Faculté des Sciences, Section 1

In this paper we address the Cauchy problem for two systems modeling the propagation of long gravity waves in a layer of homogeneous, incompressible and inviscid fluid delimited above by a free surface, and below by a non-necessarily flat rigid bottom. Concerning the Green-Naghdi system, we improve the result of Alvarez-Samaniego and Lannes (Invent. Math., 2008) in the sense that much less regular data are allowed, and no loss of derivatives is involved. Concerning the Boussinesq-Peregrine system, we improve the lower bound on the time of existence provided by Mésognon-Gireau (Adv. Differential Equations, 2017). The main ingredient is a physically motivated change of unknowns revealing the quasilinear structure of the systems, from which energy methods are implemented.

Mots clés: Boussinesq-Peregrine model; Green-Naghdi model; Initial value problems; Well-posedness theory; Energy methods; Shallow waters

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Duchêne, Samer Israwi. Well-posedness of the Green-Naghdi and Boussinesq-Peregrine systems. Annales Mathématiques Blaise Pascal, 2018, 25 (1), pp.21-74. ⟨10.5802/ambp.372⟩. ⟨hal-01396137v2⟩