American Institute of Mathematical Sciences

Advanced
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 & Heterogeneous Media, 2009, 4 (3) : 469-500. doi: 10.3934/nhm.2009.4.469
[1]

Shiping Cao, Shuangping Li, Robert S. Strichartz, Prem Talwai. A trace theorem for Sobolev spaces on the Sierpinski gasket. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3901-3916. doi: 10.3934/cpaa.2020159
[2]

Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018
[3]

Yushi Nakano, Shota Sakamoto. Spectra of expanding maps on Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1779-1797. doi: 10.3934/dcds.2019077
[4]

Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241
[5]

Baoxiang Wang. E-Besov spaces and dissipative equations. Communications on Pure & Applied Analysis, 2004, 3 (4) : 883-919. doi: 10.3934/cpaa.2004.3.883
[6]

Hermann Brunner, Jingtang Ma. Abstract cascading multigrid preconditioners in Besov spaces. Communications on Pure & Applied Analysis, 2006, 5 (2) : 349-365. doi: 10.3934/cpaa.2006.5.349
[7]

Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427
[8]

Irena Lasiecka, Buddhika Priyasad, Roberto Triggiani. Uniform stabilization of Boussinesq systems in critical $ \mathbf{L}^q $-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 4071-4117. doi: 10.3934/dcdsb.2020187
[9]

SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026
[10]

Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284
[11]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395
[12]

Houyu Jia, Xiaofeng Liu. Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces. Communications on Pure & Applied Analysis, 2008, 7 (4) : 845-852. doi: 10.3934/cpaa.2008.7.845
[13]

Mikil Foss, Joe Geisbauer. Higher differentiability in the context of Besov spaces for a class of nonlocal functionals. Evolution Equations & Control Theory, 2013, 2 (2) : 301-318. doi: 10.3934/eect.2013.2.301
[14]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[15]

Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[16]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020255
[17]

Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967
[18]

Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809
[19]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020365
[20]

Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences