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Effective interface conditions for a porous medium type problem

Giorgia Ciavolella, Noemi David, Alexandre Poulain

Abstract

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

Effective interface conditions for a porous medium type problem

Abstract

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

Paper Structure

This paper contains 10 sections, 6 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivations and previous works.
  3. Outline of the paper.
  4. Assumptions and notations
  5. A priori estimates
  6. Limit
  7. Extension operator and compactness
  8. Test function space and passage to the limit
  9. Conclusions and perspectives
  10. Existence of weak solution of the initial problem

Key Result

Theorem 1.1

Solutions of Problem epspb converge weakly to solutions $(\tilde{u},\tilde{p})$ of Problem effectivepb in the following weak form for all test functions $w(t,x)$ with a proper regularity (defined in Theorem thm: existence) and $w(T,x) = 0$ a.e. in $\Omega$. We used the notation and $(\cdot)_{|x_3=0^-}= \mathcal{T}_1 (\cdot)$ as well as $(\cdot)_{|x_3=0^+}= \mathcal{T}_3 (\cdot)$, with $\mathcal{T

Figures (2)

  • Figure 1: We represent here the bounded cylindrical domain $\Omega$ of length $L$. On the left, we can see the subdomains $\Omega_{i,\varepsilon}$ with related outward normals. The membrane $\Omega_{2,\varepsilon}$ of thickness $\varepsilon >0$ is delimited by $\Gamma_{i,i+1,\varepsilon}=\{x_3=\pm \varepsilon/2\}\cap\Omega$ which are symmetric with respect to the effective interface, $\tilde{\Gamma}_{1,3}=\{x_3=0\}\cap\Omega$. On the right, we represent the limit domain as $\varepsilon\rightarrow 0$. The effective interface, $\tilde{\Gamma}_{1,3}$, separates the two limit domains, $\tilde{\Omega}_1,\tilde{\Omega}_3$.
  • Figure 2: Representation of the spatial symmetry used in the definition of the extension operator, cf. Equation \ref{['eq: extension operator']} and of the two subdomains of $\Omega_{2,1,\varepsilon}$ and $\Omega_{2,3,\varepsilon}$.

Theorems & Definitions (11)

  • Theorem 1.1: Convergence to the effective problem
  • Lemma 3.1: A priori estimates
  • proof
  • Lemma 4.2: Compactness of the extension operator
  • proof
  • Theorem 4.3
  • proof
  • Lemma 4.4
  • proof : Proof of Lemma \ref{['lemma: intermediate']}
  • Theorem A.1: Existence of weak solutions for the initial problem
  • ...and 1 more