Silva Perez, Kevin

Profile Picture

Filters

20231
Reset filters

Settings

Citations

Publication Search Results

Now showing 1 - 1 of 1
  • Thumbnail Image
    Publication
    Domains with ramified fractal boundaries and their effects on difussion in bronchial trees
    (2023-07-04) Silva Perez, Kevin; Vélez-Santiago, Alejandro; College of Arts and Sciences - Sciences; Ríos Soto, Karen R.; Vásquez Urbano, Pedro; Department of Mathematics; Harmsen, Eric W.
    We investigate a class of domains with ramified fractal boundaries, which are an idealization of the bronchial trees in R2. Following the [53] approach, we provide a construction for these domains (Ω) with Γ∞ fractal boundary, for a parameter a with 1/2 ≤ a ≤ a∗ ≃ 0.593465. In turn, we establish several properties for the sets Ω and Γ∞, and prove in particular that Ω is a 2-set, and that Γ∞ is a d-set for d := − log (2)/ log (a). Also, motivated by some method employed by Jia [4, 5], we construct approximating sequences {an} and {bn} for the Hausdorff measure H d(Γ∞) of the fractal boundary Γ∞, in the sense that a_n ≤ H^d(Γ∞) ≤ b_n and a_n ↗ H d(Γ∞) ↙ b_n. In this regard, a way to approximate the length of bronchial trees in the pulmonary system. Next, we examine the diffusion of oxygen through the bronchial trees by considering the realization of a generalized diffusion equation ∂u/∂t − A u + Bu = f (t, x) in (0, ∞) × Ω with mixed Dirichlet-Robin boundary conditions ∂u/∂νA+ βu = g(t, x) on (0, ∞) × Γ∞; u = 0 on (0, ∞) × (Γ \ Γ∞); u(0, x) = u_0 ∈ L2(Ω), where A is an uniformly elliptic second-order (non-symmetric) differential operator with bounded measurable coefficients, B is a lower-order differential operator with unbounded measurable coefficients, and μ := ∂u/∂νA stands as a generalized notion of a normal derivative on irregular surfaces. Furthermore, β ∈ L^s_μ(Γ∞)^+ with ess inf|β(x)| ≥ β0, for some β0 > 0, x ∈ Ω and s > 1. Thus, following [37], a description of the variational formulation is presented, which is used for the qualitative study of the diffusion equation: Existence and uniqueness of the solution in the weak sense. For the adaptations of the problem: elliptic (9.2) and parabolic (10.3), the obtained results are largely related to the Lax-Milgram theorem (5.1.4) and semigroup theory (section 5.3), respectively.