Abstract
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 ofthe 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.
Highlights
The human bronchial tree can be seen as a set of dyadically connected pipes, which sums up to 23 bifurcation levels from the trachea to terminal branches, on which gas exchanges occur
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
Summary
We first gather some definitions and standard properties of 2-adic numbers (see e.g. [4]). For any z ∈ Z, z = z′2α, with z′ odd, one defines valuation v2(z) as α One extends this definition to rational numbers by setting v2(q) = v2(a) − v2(b) for any q = a/b ∈ Q, q = 0, and v2(0) = +∞. Setting |q|2 = 2−v2(q), the 2-adic distance over Q is defined as (q, q′) ∈ Q × Q −→ |q′ − q|2. For two ends x and x′, the 2-adic distance measures their proximity with regards to the tree, more precisely n = − log2 |x′ − x|2 is the index of the generation at which the corresponding paths splitted. The following proposition allows to define a trace of H1 functions over T as soon as some condition on the resistances is met : Proposition 1 Let Tr be a resistive dyadic tree, with r = (rnk ). As for the infinite version of the actual human lungs, resistances vary like αn with α ≈ 1.6, so that such a trace operator can be defined properly in Lp(Z2) for p ≤ 2.9
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