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 @e related to the depth. First we make a change of variable to a domain independent of @e and then we use asymptotic analysis to study what happens when @e becomes small. This way we obtain two new models for @e 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).