From Euler and Navier–Stokes equations to shallow waters by asymptotic analysis

In this paper, we study the Navier–Stokes and Euler 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. This way we obtain two new models for ε small that, after coming back to the original domain and without making a priori assumptions about velocity or pressure behaviour, give us a shallow water model including a new diffusion term (obtained from Navier–Stokes equations) and a shallow water model without viscosity and explicit dependence on depth (obtained from Euler equations).