Geodesic
Solutions fortes des équations de Navier-Stokes avec conditions dissipatives naturelles
C. Grandmont1 ; A. Soualah2
1 INRIA, Projet REO Rocquencourt, BP 105 78153 Le Chesnay Cedex, France
2 Laboratoire de mathématiques , Université Paris-Sud Bâtiment 425, bureau 130, 91405 Orsay Cedex, France
ESAIM. Proceedings, Tome 25 (2008), pp. 1-18.

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We are interested in the modelling of the air flow in the respiratory tract. We propose a decomposition of the respiratory tree into two stages: a proximal part, the trachea and the first generations of bronchial tubes (around the fifth or the sixth one) and a distal part, corresponding to the complex geometrical part of the bronchial tree. In the upper part, we assume that the flow is governed by the Navier-Stokes equations. The lower part is a network of tubes of low diameters and we suppose that, in each tube, the flow is described by the Poiseuille law and thus regulated by the pressure drop between the inlets and the outlets. Then each network can be condensed and replaced by a non standard boundary condition for the Navier-Stokes system. Such boundary condition is also used for heamodynamics modelling, and in this paper we provide a well posedness analysis of this multiscale model by proving the existence and uniqueness of a solution locally in time for any data, and the existence and uniqueness of solutions globally in time for small data. The continuous dependance of the solution with respect to the data is also proved. 

DOI : 10.1051/proc:082501

C. Grandmont 1 ; A. Soualah 2

1 INRIA, Projet REO Rocquencourt, BP 105 78153 Le Chesnay Cedex, France
2 Laboratoire de mathématiques , Université Paris-Sud Bâtiment 425, bureau 130, 91405 Orsay Cedex, France 
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     title = {Solutions fortes des \'equations de {Navier-Stokes} avec conditions dissipatives naturelles},
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C. Grandmont; A. Soualah. Solutions fortes des équations de Navier-Stokes avec conditions dissipatives naturelles. ESAIM. Proceedings, Tome 25 (2008), pp. 1-18. doi : 10.1051/proc:082501. http://geodesic.mathdoc.fr/articles/10.1051/proc:082501/

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