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September  2009, 4(3): 469-500. doi: 10.3934/nhm.2009.4.469

Trace theorems for trees and application to the human lungs

Bertrand Maury 1, , Delphine Salort 2, and  Christine Vannier 3,

1. 

Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex, France
2. 

Institut Jacques Monod, Université Paris Diderot, 2, Place Jussieu, 75005 Paris, France
3. 

Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France

Received  February 2008 Revised  February 2009 Published  July 2009

The aim of this paper is to develop a model of the respiratory system. The real bronchial tree is embedded within the parenchyma, and ventilation is caused by negative pressures at the alveolar level. We aim to describe the series of pressures at alveolae in the form of a function, and to establish a sound mathematical framework for the instantaneous ventilation process. To that end, we treat the bronchial tree as an infinite resistive tree, we endow the space of pressures at bifurcating nodes with the natural energy norm (rate of dissipated energy), and we characterise the pressure field at its boundary (i.e. set of simple paths to infinity). In a second step, we embed the infinite collection of leafs in a bounded domain Ω$\subset \RR^d$, and we establish some regularity properties for the corresponding pressure field. In particular, for the infinite counterpart of a regular, healthy lung, we show that the pressure field lies in a Sobolev space $H^{s}$(Ω), with $s \approx 0.45$. This allows us to propose a model for the ventilation process that takes the form of a boundary problem, where the role of the boundary is played by a full domain in the physical space, and the elliptic operator is defined over an infinite dyadic tree.
Keywords: Besov spaces., trace theorems, Sobolev spaces, Infinite resistive tree.
Mathematics Subject Classification: 05C05, 46E35, 46E3.
Citation: Bertrand Maury, Delphine Salort, Christine Vannier. Trace theorems for trees and application to the human lungs. Networks and Heterogeneous Media, 2009, 4 (3) : 469-500. doi: 10.3934/nhm.2009.4.469
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