  The purpose of this paper is to derive rigorously the so called viscous
shallow water equations given for instance page 958-959 in [A. Oron, S.H.
Davis, S.G. Bankoff, Rev. Mod. Phys, 69 (1997), 931?980]. Such a system of
equations is similar to compressible Navier-Stokes equations for a barotropic
fluid with a non-constant viscosity. To do that, we consider a layer of
incompressible and Newtonian fluid which is relatively thin, assuming no
surface tension at the free surface. The motion of the fluid is described by 3d
Navier-Stokes equations with constant viscosity and free surface. We prove that
for a set of suitable initial data (asymptotically close to "shallow water
initial data"), the Cauchy problem for these equations is well-posed, and the
solution converges to the solution of viscous shallow water equations. More
precisely, we build the solution of the full problem as a perturbation of the
strong solution to the viscous shallow water equations. The method of proof is
based on a Lagrangian change of variable that fixes the fluid domain and we
have to prove the well-posedness in thin domains: we have to pay a special
attention to constants in classical Sobolev inequalities and regularity in
Stokes problem.
