Derivation and analysis of a new 2D Green-Naghdi system

Samer Israwi (1)

1 IMB - Institut de Mathématiques de Bordeaux

We derive here a variant of the $2D$ Green-Naghdi equations that model the propagation of two-directional, nonlinear dispersive waves in shallow water. This new model has the same accuracy as the standard $2D $ Green-Naghdi equations. Its mathematical interest is that it allows a control of the rotational part of the (vertically averaged) horizontal velocity, which is not the case for the usual Green-Naghdi equations. Using this property, we show that the solution of these new equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition. Finally, we prove that the new Green-Naghdi equations conserve the almost irrotationality of the vertically averaged horizontal component of the velocity.

Mots clés: Green-Naghdi equations; water-waves

Domaines: Equations aux dérivées partielles [math.AP]

Samer Israwi. Derivation and analysis of a new 2D Green-Naghdi system. 2009. ⟨hal-00447005⟩