Mathematical justification of a shallow water model. (English) Zbl 1158.35401

Bresch, Didier; Noble, Pascal

Methods Appl. Anal. 14, No. 2, 87-117 (2007).

Summary: The shallow water equations are widely used to model the flow of a thin layer of fluid submitted to gravity forces. They are usually formally derived from the full incompressible Navier-Stokes equations with free surface under the modeling hypothesis that the pressure is hydrostatic, the flow is laminar, gradually varied and the characteristic fluid height is small relative to the characteristics flow length. This paper deals with the mathematical justification of such asymptotic process assuming a non zero surface tension coefficient and some constraints on the data. We also discuss relation between lubrication models and shallow water systems with no surface tension coefficient necessity.

MSC:

35Q30	Navier-Stokes equations
35R35	Free boundary problems for PDEs
76A20	Thin fluid films
76B45	Capillarity (surface tension) for incompressible inviscid fluids
76D08	Lubrication theory

Keywords:

Navier-Stokes equations; shallow water; lubrication models; thin domain; free surface; asymptotic analysis; Sobolev spaces