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Emergence of biological transportation networks as a self-regulated process

Jan Haskovec, Peter Markowich, Simone Portaro

TL;DR

A formal gradient flow for the symmetric tensor valued diffusivity of a broad class of entropy dissipations associated with a purely diffusive model is written, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs.

Abstract

We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal $L^2$-gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs.

Emergence of biological transportation networks as a self-regulated process

TL;DR

A formal gradient flow for the symmetric tensor valued diffusivity of a broad class of entropy dissipations associated with a purely diffusive model is written, giving an evolution equation for coupled with two auxiliary elliptic PDEs.

Abstract

We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal -gradient flow for the symmetric tensor valued diffusivity of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal -gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for coupled with two auxiliary elliptic PDEs.
Paper Structure (14 sections, 9 theorems, 68 equations)

This paper contains 14 sections, 9 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. D
  3. D
  4. D
  5. General diffusive model
  6. Drift-diffusion model
  7. D
  8. D
  9. w
  10. w^0
  11. Poisson-Nernst-Planck model
  12. u
  13. u
  14. Conclusions

Key Result

Lemma 1

The formal $L^2$-gradient flow of the energy functional eq:energy_genmod constrained by eq:ell, BC_u is given by for the symmetric tensor valued diffusivity $D=D(t,x)$ and scalar $u=u(t,x)$, with $\sigma=\sigma(t,x)$ the solution of the boundary value problem subject to $\sigma = 0$ on $\partial \Omega$.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Lemma 4
  • proof
  • ...and 8 more