Derivation of a viscous KP equation including surface tension, and related equations

The aim of this article is to derive asymptotic models from surface wave equations in the presence of surface tension and viscosity. Using the Navier-Stokes equations with a flat bottom, we derive the viscous 2D Boussinesq system. The assumed scale of transverse variation is larger than the one along the main propagation direction (weak transverse variation). This Boussinesq system is proved to be consistent with the Navier-Stokes equations. This system is only an intermediate result that enables us to derive the Kadomtsev-Petviashvili (KP) equation which is a 2D generalization of the KdV equation. In addition, we get the 1D KdV equation, and lastly the Boussinesq equation. All these equations are derived for general initial conditions either slipping (Euler's fluid) or sticking (Navier-Stokes fluid) with a given profile in the boundary layer different from the Euler's one. We discuss whether the Euler's initial condition is physical.