Models for Damped Water Waves

Abstract

In this paper we derive some new weakly nonlinear asymptotic models describing viscous waves in deep water with or without surface tension effects. These asymptotic models take into account several different dissipative effects and are obtained from the free boundary problems formulated in the works of Dias, Dyachenko, and Zakharov [Phys.

Asymptotic models for free boundary flow in porous media

“…(1.1) Equation (AD ν ) was derived by the authors as an asymptotic model for the Darcy flow in porous media under the assumption that the amplitude over the wavelength, a quotient known as steepness, is small [27] (see also [9,28,30]).…”

Interfaces in incompressible flows

“…To have a system of PDEs that includes both approaches, we define the following free boundary problem [28]:…”

“…Although we will focus in models for the case of Euler and Navier-Stokes equations, similar models have been obtained and studied for waves in porous media in [27,29]. We start with the models for waves with small steepness and viscosity in [25,26,28,30,31]. In particular, in section 2 we present the models for viscous fluids with shear viscosiy in [26,28,30,31] and the models for viscous fluids with odd or Hall viscosity in [25].…”

“…We start with the models for waves with small steepness and viscosity in [25,26,28,30,31]. In particular, in section 2 we present the models for viscous fluids with shear viscosiy in [26,28,30,31] and the models for viscous fluids with odd or Hall viscosity in [25]. In section 3 we review the models in [14] for inviscid fluids.…”

“…These models take the form of both bidirectional and unidirectional nonlinear and nonlocal wave equations. More precisely, with the goal of telling a more complete story, in this paper we present the results in the works [14,25,26,28,[30][31][32] together with some new results regarding the wellposedness of the resulting PDEs. Contents 1.…”