# American Institute of Mathematical Sciences

March  2007, 7(2): 219-250. doi: 10.3934/dcdsb.2007.7.219

## Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations

 1 CNRS / Université de Provence, Laboratoire d’Analyse, Topologie et Probabilités, UMR 6632, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France 2 Université Bordeaux-I, Mathématiques Appliquées, 351 Cours de la Libération, 33405 Talence Cedex

Received  January 2006 Revised  September 2006 Published  December 2006

In this paper we propose the analysis of the incompressible non-homogeneous Navier-Stokes equations with nonlinear outflow boundary condition. This kind of boundary condition appears to be, in some situations, a useful way to perform numerical computations of the solution to the unsteady Navier-Stokes equations when the Dirichlet data are not given explicitly by the physical context on a part of the boundary of the computational domain. The boundary condition we propose, following previous works in the homogeneous case, is a relationship between the normal component of the stress and the outflow momentum flux taking into account inertial effects. We prove the global existence of a weak solution to this model both in 2D and 3D. In particular, we show that the nonlinear boundary condition under study holds for such a solution in a weak sense, even though the normal component of the stress and the density may not have traces in the usual sense.
Citation: Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219
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