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Analysis of a degenerate parabolic system for cell dynamics in intestinal crypts

Ahmad El Hajj, Mohamad El Hajj Chehade, Antoine Zurek

TL;DR

This work studies a system of degenerate parabolic equations modeling the dynamics of multiple cell populations in intestinal crypts and considers a regularized form of the system and establishes uniform BV estimates.

Abstract

In this work, we study a system of degenerate parabolic equations modeling the dynamics of multiple cell populations in intestinal crypts. The model describes cell division, differentiation, and migration through a strongly coupled system of reaction-cross-diffusion equations with degenerate diffusion. By working with initial data in BV, we first consider a regularized form of the system and establish uniform BV estimates. Using these bounds, we then pass to the limit to obtain the existence of weak solutions.

Analysis of a degenerate parabolic system for cell dynamics in intestinal crypts

TL;DR

This work studies a system of degenerate parabolic equations modeling the dynamics of multiple cell populations in intestinal crypts and considers a regularized form of the system and establishes uniform BV estimates.

Abstract

In this work, we study a system of degenerate parabolic equations modeling the dynamics of multiple cell populations in intestinal crypts. The model describes cell division, differentiation, and migration through a strongly coupled system of reaction-cross-diffusion equations with degenerate diffusion. By working with initial data in BV, we first consider a regularized form of the system and establish uniform BV estimates. Using these bounds, we then pass to the limit to obtain the existence of weak solutions.
Paper Structure (14 sections, 6 theorems, 108 equations, 1 figure)

This paper contains 14 sections, 6 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Mathematical model and main result
  4. Introduction of the mathematical model
  5. Mathematical difficulties and related studies
  6. Assumptions and main result
  7. Study of a regularized system
  8. Study of the equation on the total density
  9. Study of the equations on the partial densities
  10. Study of the equation on the concentration
  11. Proof of Proposition \ref{['prop : semi discrete']}
  12. Existence proof
  13. Uniform estimates
  14. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 2.1

Let the assumptions (H1)-- (H4) hold. Then, there exist nonnegative functions $\rho_{{\rm s}}$, $\rho_{{\rm p}}$, $\rho_{{\rm e}}$ and $\rho_{{\rm g}}$ such that $\rho_i \in L^\infty(0,T;BV(\Omega))$ with $\partial_t \rho_i \in L^2(0,T;(H^2)'(\Omega))$ for any $i \in \mathcal{T}$ and $\rho\in L^2(0 and for all $\psi \in L^2(0,T;H)$ where we recall definition def : H of the space $H$.

Figures (1)

  • Figure 1: Schematic representation of the repartition of the cell colonies inside the colonic crypt.

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Theorem 2.1: Existence of weak solutions
  • Proposition 1
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 3 more