A 2-adic approach of the human respiratory tree

We propose here a general framework to address the question of traceoperators on a dyadic tree. This work is motivated by the modeling of the humanbronchial tree which, thanks to its regularity, can be extrapolated in anatural way to an infinite resistive tree. The space of pressure fields atbifurcation nodes of this infinite tree can be endowed with a Sobolev spacestructure, with a semi-norm which measures the instantaneous rate of dissipatedenergy. We aim at describing the behaviour of finite energy pressure fieldsnear the end. The core of the present approach is an identification of the setof ends with the ring Z_2 of 2-adic integers. Sobolev spaces over Z_2 can bedefined in a very natural way by means of Fourier transform, which allows us toestablish precised trace theorems which are formally quite similar to those instandard Sobolev spaces, with a Sobolev regularity which depends on the growthrate of resistances, i.e. on geometrical properties of the tree. Furthermore,we exhibit an explicit expression of the "ventilation operator", which mapspressure fields at the end of the tree onto fluxes, in the form of aconvolution by a Riesz kernel based on the 2-adic distance.