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Derivation and analysis of a Stokes-transport system in evolving vessels modeling thermoregulation in human skin

Kilian Hacker, Maria Neuss-Radu

TL;DR

The article develops a rigorous, fully coupled Stokes-transport-ODE model for thermoregulation in skin, featuring an evolving vessel-domain connected to heat transfer and NO-driven vasomotion. It transforms the problem to a reference domain and proves global-in-time existence and uniqueness of a weak solution using a Schaefer fixed-point framework, handling nonlinear couplings through careful estimates and operator continuity. Key contributions include a deformable-domain formulation with transmission heat conditions, a Galerkin treatment of the advection-diffusion part, and a comprehensive set of a priori estimates ensuring well-posedness of the fully coupled system. This framework provides a mathematically solid basis for analyzing thermoregulatory feedback in skin and informs potential numerical schemes for local vascular heat-control modeling.

Abstract

We consider a Stokes flow coupled with advective-diffusive transport in an evolving domain with boundary conditions allowing for inflow and outflow. The evolution of the domain is induced by the transport process, leading to a fully coupled problem. Our aim is to model the thermal control of blood flow in human skin. To this end, the model takes into account the temperature-dependent production of biochemical substances, the subsequent dilation and constriction of blood vessels, and the resulting changes in convective heat transfer. We prove existence and uniqueness of weak solutions using a fixed point method that allows us to treat the nonlinear coupling.

Derivation and analysis of a Stokes-transport system in evolving vessels modeling thermoregulation in human skin

TL;DR

The article develops a rigorous, fully coupled Stokes-transport-ODE model for thermoregulation in skin, featuring an evolving vessel-domain connected to heat transfer and NO-driven vasomotion. It transforms the problem to a reference domain and proves global-in-time existence and uniqueness of a weak solution using a Schaefer fixed-point framework, handling nonlinear couplings through careful estimates and operator continuity. Key contributions include a deformable-domain formulation with transmission heat conditions, a Galerkin treatment of the advection-diffusion part, and a comprehensive set of a priori estimates ensuring well-posedness of the fully coupled system. This framework provides a mathematically solid basis for analyzing thermoregulatory feedback in skin and informs potential numerical schemes for local vascular heat-control modeling.

Abstract

We consider a Stokes flow coupled with advective-diffusive transport in an evolving domain with boundary conditions allowing for inflow and outflow. The evolution of the domain is induced by the transport process, leading to a fully coupled problem. Our aim is to model the thermal control of blood flow in human skin. To this end, the model takes into account the temperature-dependent production of biochemical substances, the subsequent dilation and constriction of blood vessels, and the resulting changes in convective heat transfer. We prove existence and uniqueness of weak solutions using a fixed point method that allows us to treat the nonlinear coupling.
Paper Structure (15 sections, 11 theorems, 135 equations, 2 figures)

This paper contains 15 sections, 11 theorems, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Mathematical model
  4. The evolving domain
  5. Stokes system modeling blood flow
  6. Advection-diffusion equation modeling temperature evolution
  7. ODE modeling chemical reactions
  8. Weak formulation in the reference domain
  9. Global-in-time existence and uniqueness
  10. Existence of a unique solution for the ODE
  11. Deformation of the domain and its properties
  12. Existence of a unique solution for the Stokes system
  13. Existence of a unique solution for the advection-diffusion equation
  14. Existence of solutions for the fully coupled system
  15. Uniqueness of solutions for the fully coupled system

Key Result

Theorem 3.1

Under the Assumption assu:data, there exists a unique weak solution $(c,w,q,\vartheta_f,\vartheta_s)$ to the fully coupled system eq:solution_spaces--eq:coefficients.

Figures (2)

  • Figure 1: The evolving domain (left) and the reference domain (right), both divided into a fluid and a solid part.
  • Figure 2: The fully coupled system of equations (left) and the linearized system (right).

Theorems & Definitions (28)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.6
  • Theorem 3.1: Global-in-time existence and uniqueness
  • Lemma 3.2
  • Proof
  • Proposition 3.3
  • Proof
  • ...and 18 more