From Navier–Stokes equations to shallow waters with viscosity by asymptotic analysis

Abstract

In this paper, we study the Navier–Stokes equations in a domain with small depth. With this aim, we introduce a small adimensional parameter ε related to the depth. First we make a change of variable to a domain independent of ε and then we use asymptotic analysis to study what happens when ε becomes small. In this way we obtain a model for ε small that, after coming back to the original domain and without making a priori assumptions about velocity or pressure behavior, gives us a shallow water model including a new diffusion term.