September  2010, 5(3): 405-422. doi: 10.3934/nhm.2010.5.405

A 2-adic approach of the human respiratory tree

1. 

Laboratoire Paul Painlevé - CNRS, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France

2. 

Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex

3. 

Institut Jacques Monod, Université Paris Diderot, Bât. Buffon, 15 rue Hélène Brion, 75013 Paris, France

Received  January 2010 Revised  April 2010 Published  July 2010

We propose here a general framework to address the question of trace operators on a dyadic tree. This work is motivated by the modeling of the human bronchial tree which, thanks to its regularity, can be extrapolated in a natural way to an infinite resistive tree. The space of pressure fields at bifurcation nodes of this infinite tree can be endowed with a Sobolev space structure, with a semi-norm which measures the instantaneous rate of dissipated energy. We aim at describing the behaviour of finite energy pressure fields near the end. The core of the present approach is an identification of the set of ends with the ring $\ZZ$2 of 2-adic integers. Sobolev spaces over $\ZZ$2 can be defined in a very natural way by means of Fourier transform, which allows us to establish precised trace theorems which are formally quite similar to those in standard Sobolev spaces, with a Sobolev regularity which depends on the growth rate of resistances, i.e. on geometrical properties of the tree. Furthermore, we exhibit an explicit expression of the "ventilation operator'', which maps pressure fields at the end of the tree onto fluxes, in the form of a convolution by a Riesz kernel based on the 2-adic distance.
Citation: Frédéric Bernicot, Bertrand Maury, Delphine Salort. A 2-adic approach of the human respiratory tree. Networks and Heterogeneous Media, 2010, 5 (3) : 405-422. doi: 10.3934/nhm.2010.5.405
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