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.
"On the long time behaviour of the solution to the two-fluids incompressible Navier-Stokes equations." Differential Integral Equations 12 (5) 691 - 740, 1999.