## Methods and Applications of Analysis

- Methods Appl. Anal.
- Volume 14, Number 2 (2007), 87-118.

### Mathematical Justification of a Shallow Water Model

Didier Bresch and Pascal Noble

#### Abstract

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.

#### Article information

**Source**

Methods Appl. Anal., Volume 14, Number 2 (2007), 87-118.

**Dates**

First available in Project Euclid: 7 July 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.maa/1215442819

**Mathematical Reviews number (MathSciNet)**

MR2437099

**Zentralblatt MATH identifier**

1158.35401

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35R35: Free boundary problems 76A20: Thin fluid films 76B45: Capillarity (surface tension) [See also 76D45] 76D08: Lubrication theory

**Keywords**

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

#### Citation

Bresch, Didier; Noble, Pascal. Mathematical Justification of a Shallow Water Model. Methods Appl. Anal. 14 (2007), no. 2, 87--118. https://projecteuclid.org/euclid.maa/1215442819