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 [22]. In this context, we answer positively the question raised by [12] for the derivation and thus complete the derivation.
Accepté le :
Publié le :
Helge Dietert 1 ; Ayman Moussa 2
![](/static/ptf/img/CC-BY%204.0.png)
@article{AFST_2024_6_33_1_69_0, author = {Helge Dietert and Ayman Moussa}, title = {Persisting entropy structure for nonlocal cross-diffusion systems}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {69--104}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 33}, number = {1}, year = {2024}, doi = {10.5802/afst.1762}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1762/} }
TY - JOUR AU - Helge Dietert AU - Ayman Moussa TI - Persisting entropy structure for nonlocal cross-diffusion systems JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2024 SP - 69 EP - 104 VL - 33 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1762/ DO - 10.5802/afst.1762 LA - en ID - AFST_2024_6_33_1_69_0 ER -
%0 Journal Article %A Helge Dietert %A Ayman Moussa %T Persisting entropy structure for nonlocal cross-diffusion systems %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2024 %P 69-104 %V 33 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1762/ %R 10.5802/afst.1762 %G en %F AFST_2024_6_33_1_69_0
Helge Dietert; Ayman Moussa. Persisting entropy structure for nonlocal cross-diffusion systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 1, pp. 69-104. doi : 10.5802/afst.1762. https://afst.centre-mersenne.org/articles/10.5802/afst.1762/
[1] Erratum: “Dynamic theory of quasilinear parabolic systems. III. Global existence” [Math. Z. 202 (1989), no. 2, 219–250; MR1013086 (90i:35125)], Math. Z., Volume 205 (1990) no. 2, p. 331 | DOI | MR | Zbl
[2] Stability of a cross-diffusion system and approximation by repulsive random walks: a duality approach (2021) | arXiv
[3] Conservative cross diffusions and pattern formation through relaxation, J. Math. Pures Appl., Volume 92 (2009) no. 6, pp. 651-667 http://www.sciencedirect.com/science/article/pii/s0021782409000592 | DOI | Zbl
[4] Local Minimization, Variational Evolution and -Convergence, Lecture Notes in Mathematics, Springer, 2014, xi+174 pages | DOI
[5] Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems, J. Nonlinear Sci., Volume 31 (2021) no. 6, 94, 38 pages | DOI | Zbl
[6] Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differ. Equations, Volume 224 (2006) no. 1, pp. 39-59 | DOI | MR | Zbl
[7] Global existence analysis of cross-diffusion population systems for multiple species, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 2, pp. 715-747 | DOI | MR | Zbl
[8] About the entropic structure of detailed balanced multi-species cross-diffusion equations, J. Differ. Equations, Volume 266 (2019) no. 7, pp. 3861-3882 | DOI | Zbl
[9] Cross-diffusion systems and fast-reaction limits, Bull. Sci. Math., Volume 159 (2020), 102824, 29 pages | DOI | Zbl
[10] On the entropic structure of reaction-cross diffusion systems, Commun. Partial Differ. Equations, Volume 40 (2015) no. 9, pp. 1705-1747 | DOI | MR | Zbl
[11] Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 2010 | DOI
[12] Non local Lotka-Volterra system with cross-diffusion in an heterogeneous medium, J. Math. Biol., Volume 70 (2014) no. 4, pp. 829-854 | DOI | Zbl
[13] Gradient Estimates and Global Existence of Smooth Solutions to a Cross-Diffusion System, SIAM J. Math. Anal., Volume 47 (2015) no. 3, pp. 2122-2177 | DOI | Zbl
[14] Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., Volume 53 (2006) no. 4, pp. 617-641 | DOI | Zbl
[15] The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, Volume 28 (2015) no. 6, pp. 1963-2001 | DOI | Zbl
[16] Entropic structure and duality for multiple species cross-diffusion systems, Nonlinear Anal., Theory Methods Appl., Volume 159 (2017), pp. 298-315 | DOI | MR | Zbl
[17] Global well-posedness of a conservative relaxed cross diffusion system, SIAM J. Math. Anal., Volume 44 (2012) no. 3, pp. 1674-1693 | DOI | MR | Zbl
[18] On Evolutionary -Convergence for Gradient Systems, Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity (Lecture Notes in Applied Mathematics and Mechanics), Volume 3, Springer, 2016, pp. 187-249 | DOI
[19] Some variants of the classical Aubin-Lions lemma, J. Evol. Equ., Volume 16 (2016) no. 1, pp. 65-93 | DOI | MR | Zbl
[20] From Nonlocal to Classical Shigesada–Kawasaki–Teramoto Systems: Triangular Case with Bounded Coefficients, SIAM J. Math. Anal., Volume 52 (2020) no. 1, pp. 42-64 | DOI | Zbl
[21] Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., Ser. A, Volume 31 (2011) no. 4, pp. 1427-1451 | DOI | Zbl
[22] Spatial segregation of interacting species, J. Theor. Biol., Volume 79 (1979) no. 1, pp. 83-99 | DOI
[23] Singular Integrals and Differentiability Properties of Functions (PMS-30), Princeton University Press, 1970 | DOI
[24] On triangular reaction cross-diffusion systems with possible self-diffusion, Bull. Sci. Math., Volume 140 (2016) no. 7, pp. 796-829 | DOI | Zbl
[25] Transport distances and geodesic convexity for systems of degenerate diffusion equations, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 4, pp. 3397-3438 | DOI | Zbl
Cité par Sources :