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Stability of a cross-diffusion system and approximation by repulsive random walks: a duality approach

Abstract : We consider conservative cross-diffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first control the time evolution of the gap between two bounded solutions by means of its initial value. As a by product, we obtain a uniqueness result for bounded solutions valid for any space dimension, under a smallness assumption. Using a discrete counterpart of our duality estimates, we prove the convergence of random walks with local repulsion in one dimensional discrete space to cross-diffusion systems. More precisely, we prove sharp quantitative estimates for the gap between the stochastic process and the cross-diffusion system. We complete this study with a rough but general estimate and convergence results, when the population and the number of sites become large.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03344301
Contributor : Felipe Muñoz-Hernández Connect in order to contact the contributor
Submitted on : Tuesday, September 14, 2021 - 8:01:38 PM
Last modification on : Tuesday, September 28, 2021 - 5:16:44 PM

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  • HAL Id : hal-03344301, version 1
  • ARXIV : 2109.07146

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Vincent Bansaye, Ayman Moussa, Felipe Muñoz-Hernández. Stability of a cross-diffusion system and approximation by repulsive random walks: a duality approach. 2021. ⟨hal-03344301⟩

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