## Differential and Integral Equations

- Differential Integral Equations
- Volume 12, Number 5 (1999), 691-740.

### On the long time behaviour of the solution to the two-fluids incompressible Navier-Stokes equations

#### Abstract

We devote this work to the long time behaviour of the solution to the incompressible Navier-Stokes equations for two viscous immiscible fluids contained in a bounded domain and subjected only to gravity forces. When there is surface tension at the interface or not, for the model linearized around the steady-state of minimal energy or for the standard nonlinear model, we investigate the following question. Do the equations reproduce the behaviour expected from experiment, namely a convergence to zero of the velocity field, and a convergence of the interface to its stable position. Our results show a wide variety of behaviours, depending on the case considered.

#### Article information

**Source**

Differential Integral Equations, Volume 12, Number 5 (1999), 691-740.

**Dates**

First available in Project Euclid: 29 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367255391

**Mathematical Reviews number (MathSciNet)**

MR1697251

**Zentralblatt MATH identifier**

1022.35038

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Secondary: 76D05: Navier-Stokes equations [See also 35Q30]

#### Citation

Gerbeau, J.-F.; Le Bris, C. On the long time behaviour of the solution to the two-fluids incompressible Navier-Stokes equations. Differential Integral Equations 12 (1999), no. 5, 691--740. https://projecteuclid.org/euclid.die/1367255391