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Existence of global weak solutions to a Cahn-Hilliard cross-diffusion system in lymphangiogenesis

Ansgar Jüngel, Yue Li

Abstract

The global-in-time existence of weak solutions to a degenerate Cahn-Hilliard cross-diffusion system with singular potential in a bounded domain with no-flux boundary conditions is proved. The model consists of two coupled parabolic fourth-order partial differential equations and describes the evolution of the fiber phase volume fraction and the solute concentration, modeling the pre-patterning of lymphatic vessel morphology. The fiber phase fraction satisfies the segregation property if this holds initially. The existence proof is based on a three-level approximation scheme and a priori estimates coming from the energy and entropy inequalities. While the free energy is nonincreasing in time, the entropy is only bounded because of the cross-diffusion coupling.

Existence of global weak solutions to a Cahn-Hilliard cross-diffusion system in lymphangiogenesis

Abstract

The global-in-time existence of weak solutions to a degenerate Cahn-Hilliard cross-diffusion system with singular potential in a bounded domain with no-flux boundary conditions is proved. The model consists of two coupled parabolic fourth-order partial differential equations and describes the evolution of the fiber phase volume fraction and the solute concentration, modeling the pre-patterning of lymphatic vessel morphology. The fiber phase fraction satisfies the segregation property if this holds initially. The existence proof is based on a three-level approximation scheme and a priori estimates coming from the energy and entropy inequalities. While the free energy is nonincreasing in time, the entropy is only bounded because of the cross-diffusion coupling.
Paper Structure (15 sections, 12 theorems, 132 equations)

This paper contains 15 sections, 12 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. State of the art
  3. Key ideas
  4. Main result
  5. Formal estimations
  6. Approximate solutions
  7. Faedo--Galerkin approximation
  8. Uniform estimates independent of $N$
  9. Compactness argument
  10. The limit $\varepsilon\rightarrow 0$
  11. Uniform estimates independent of $\varepsilon$
  12. Compactness argument
  13. The limit $\delta\rightarrow 0$
  14. Uniform estimates independent of $\delta$
  15. Compactness argument

Key Result

Theorem 1.2

Let $\Omega\subset{\mathbb R}^d$$(d\le 4)$ be a bounded domain with boundary $\partial\Omega\in C^2$, let $T>0$, and let $\phi_0\in H^1(\Omega)$, $c_0\in L^2(\Omega)$ satisfy $0<\phi_*\le\phi_0\le 1-\phi_*<1$, $c_0\ge 0$ in $\Omega$ for some $\phi_*\in(0,1)$. Then problem m1--bc possesses a weak sol

Theorems & Definitions (24)

  • Definition 1.1: Weak solution
  • Theorem 1.2
  • Proposition 2.1: Energy equality
  • proof
  • Proposition 2.2: Entropy equality
  • proof
  • Lemma 3.1: Energy inequality
  • proof
  • Lemma 3.2
  • proof
  • ...and 14 more