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

In this paper, we study the Navier-Stokes equations in a domain with small depth. With this aim, we introduce a small adimensional parameter epsilon related to the depth. First we make a change of variable to a domain independent of epsilon and then we use asymptotic analysis to study what happens when epsilon becomes small. In this way we obtain a model for epsilon 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.