IUMJ

Title: Mathematical derivation of viscous shallow-water equations with zero surface tension

Authors: Didier Bresch and Pascal Noble

Issue: Volume 60 (2011), Issue 4, 1137-1170

Abstract:

The purpose of this paper is to derive rigorously the so-called viscous shallow-water equations given for instance in [{\sc A. Oron, S.H. Davis, S.G. Bankoff}, \textit{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 \emph{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" close to rest state), 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 of the viscous shallow-water equations. The method of proof is based on a Lagrangian change of variable which fixes the fluid domain, and we have to prove the well-posedness in thin domains: in particular, we have to pay special attention to constants in classical Sobolev inequalities and regularity in the Stokes problem.