About the entropic structure of detailed balanced multi-species cross-diffusion equations

Daus, Esther S.; Desvillettes, Laurent; Dietert, Helge

doi:10.1016/j.jde.2018.09.020

Cited by 19 publications

(35 citation statements)

References 22 publications

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“…In this work we aim at giving sufficient conditions on the mollifying process for the entropy structure to persist on the nonlocal system. Our intuition takes its origin from the article [7] in which the first author of the current article exhibited an entropy structure for the SKT systems under a spatial discretization. It is a striking fact that this very discretization is everything but a nonlocal version of the SKT system.…”

Section: Dt ωmentioning

confidence: 99%

Persisting entropy structure for nonlocal cross-diffusion systems

Dietert¹,

Moussa²

2021

Preprint

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For cross-diffusion systems possessing an entropy (i.e. a Lyapunov functional) we study nonlocal versions and exhibit sufficient conditions to ensure that the nonlocal version inherits the entropy structure. These nonlocal systems can be understood as population models per se or as approximation of the classical ones. With the preserved entropy, we can rigorously link the approximating nonlocal version to the classical local system. From a modelling perspective this gives a way to prove a derivation of the model and from a PDE perspective this provides a regularisation scheme to prove the existence of solutions. A guiding example is the SKT model [19] and in this context we answer positively to a question raised by Fontbona and Méléard in [10] and thus provide a full derivation.

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Section: Dt ωmentioning

confidence: 99%

Persisting entropy structure for nonlocal cross-diffusion systems

Dietert¹,

Moussa²

2021

Preprint

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“…Moussa [20] then proved the limit from the nonlocal to the local diffusion system (but only for triangular diffusion matrices), which gives the Shigesada-Kawasaki-Teramoto crossdiffusion system. A derivation of a space discretized version of this system from a Markov chain model was presented in [7]. Another nonlocal mean-field model was analyzed in [3].…”

Section: 2mentioning

confidence: 99%

Rigorous mean-field limit and cross-diffusion

Chen

Daus

Jüngel

2019

Z. Angew. Math. Phys.

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The mean-field limit in a weakly interacting stochastic many-particle system for multiple population species in the whole space is proved. The limiting system consists of cross-diffusion equations, modeling the segregation of populations. The mean-field limit is performed in two steps: First, the many-particle system leads in the large population limit to an intermediate nonlocal diffusion system. The local cross-diffusion system is then obtained from the nonlocal system when the interaction potentials approach the Dirac delta distribution. The global existence of the limiting and the intermediate diffusion systems is shown for small initial data, and an error estimate is given.

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“…The many-particle limit from a particle system driven by Lévy noise to a fractional cross-diffusion system related to (2) was recently shown in Daus et al (2020). Furthermore, the population system (1) was derived in Daus et al (2019) from a time-continuous Markov chain model using the BBGKY hierarchy. This paper presents, up to our knowledge, the first rigorous derivation of the Shigesada-Kawasaki-Teramoto (SKT) model (1) from a stochastic particle system in the moderate many-particle limit.…”

Section: Introductionmentioning

confidence: 99%

“…Compared to the work (Daus et al 2019), we take the limits N → ∞, η → 0 simultaneously. However, our approach also implies the two-step limit.…”

Section: Introductionmentioning

confidence: 99%

Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

et al. 2021

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Population cross-diffusion systems of Shigesada–Kawasaki–Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.

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About the entropic structure of detailed balanced multi-species cross-diffusion equations

Cited by 19 publications

References 22 publications

Persisting entropy structure for nonlocal cross-diffusion systems

Persisting entropy structure for nonlocal cross-diffusion systems

Rigorous mean-field limit and cross-diffusion

Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

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