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

TL;DR

This paper rigorously derives viscous shallow water equations from 3D Navier-Stokes equations for a thin, incompressible Newtonian fluid without surface tension, proving well-posedness and convergence of solutions.

Contribution

It provides a rigorous mathematical derivation and proof of well-posedness for viscous shallow water equations from fundamental Navier-Stokes equations in thin domains.

Findings

Proves well-posedness of the viscous shallow water equations

Shows convergence of 3D Navier-Stokes solutions to shallow water solutions

Develops a method using Lagrangian change of variables in thin domains

Abstract

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.