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On a cross-diffusion segregation problem arising from a model of interacting particles

Gonzalo Galiano, Virginia Selgas

Abstract

We prove the existence of solutions of a cross-diffusion parabolic population problem. The system of partial differential equations is deduced as the limit equations satisfied by the densities corresponding to an interacting particles system modeled by stochastic differential equations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. For proving the existence of solutions when the matrix is positive definite, we use a fully discrete finite element approximation in a general functional setting. If the matrix is only positive semi-definite, we use a regularization technique based on a related cross-diffusion model under more restrictive functional assumptions. We provide some numerical experiments demonstrating the weak and strong segregation effects corresponding to both types of matrices.

On a cross-diffusion segregation problem arising from a model of interacting particles

Abstract

We prove the existence of solutions of a cross-diffusion parabolic population problem. The system of partial differential equations is deduced as the limit equations satisfied by the densities corresponding to an interacting particles system modeled by stochastic differential equations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. For proving the existence of solutions when the matrix is positive definite, we use a fully discrete finite element approximation in a general functional setting. If the matrix is only positive semi-definite, we use a regularization technique based on a related cross-diffusion model under more restrictive functional assumptions. We provide some numerical experiments demonstrating the weak and strong segregation effects corresponding to both types of matrices.

Paper Structure

This paper contains 12 sections, 5 theorems, 95 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Mathematical modeling
  3. The Euler description
  4. Assumptions and main results
  5. Approximated problems and numerical experiments
  6. Proofs of the theorems
  7. Proof of Theorem \ref{['th:existence_original']}
  8. Finite element approximation
  9. The discrete problem
  10. Passing to the limits
  11. Proof of Theorem \ref{['th:existence_particular']}
  12. Conclusion

Key Result

Theorem 1

Let $T>0$ and assume (H). Then problem (eq:pde)-(eq:id) has a weak solution $(u_1,u_2)$ satisfying $u_i\geq 0$ in $Q_T$ and where $p=(2m+2)/(2m+1)$, $r=2(m+1)/m$ and $p'=2(m+1)$, in the sense that for all $\varphi\in L^{p'}(0,T;W^{1,p'}(\Omega))$, $i=1,2$, with $<\cdot,\cdot>$ denoting the duality product between $W^{1,p'}(\Omega)$ and its dual $(W^{1,p'}(\Omega))'$.

Figures (6)

  • Figure 1: Experiment 1. Numerical results for several values of $b_1$ using Shigesada et al. model (left panel) and the model studied in this article (right panel).
  • Figure 2: Experiment 2. The case of semi-definite positive matrix $(a_{ij})$
  • Figure 3: Experiment 2. Regularizing parameter $\delta=0.001$.
  • Figure 4: Experiment 2. Zoom into the supports intersection region.
  • Figure 5: Experiment 3. Evolution of the invasion.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3