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

Didier Bresch (1) , Pascal Noble (2)

1 LAMA - Laboratoire de Mathématiques
2 ICJ - Institut Camille Jordan

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.

Mots clés: Sobolev spaces; Navier-Stokes; shallow water; thin domain; free surface; asymptotic analysis; Sobolev spaces.

Domaines: Equations aux dérivées partielles [math.AP]

Didier Bresch, Pascal Noble. Mathematical derivation of viscous shallow-water equations with zero surface tension. 2010. ⟨hal-00456181⟩