Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems
Tóm tắt
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.
Tài liệu tham khảo
Bertsch, M., Gurtin, M., Hilhorst, D., Peletier, L.: On interacting populations that disperse to avoid crowding: preservation of segregation. J. Math. Biol. 23, 1–13 (1985)
Burger, M., Carrillo, J.A., Pietschmann, J.-F., Schmidtchen, M.: Segregation effects and gap formation in cross-diffusion models. Interfaces Free Bound. 22, 175–203 (2020a)
Burger, M., Pietschmann, J.-F., Ranetbauer, H., Schmeiser, C., Wolfram, M.-T.: Mean-field models for segregation dynamics. Eur. J. Appl. Math. To appear in arXiv:1808.04069 (2020b)
Chen, L., Göttlich, S., Knapp, S.: Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation. ESAIM: Math. Mod. Num. Anal. 53, 567–593 (2018a)
Chen, X., Daus, E.S., Jüngel, A.: Global existence analysis of cross-diffusion population systems for multiple species. Arch. Ration. Mech. Anal. 227, 715–747 (2018b)
Chen, L., Daus, E.S., Jüngel, A.: Rigorous mean-field limits and cross diffusion. Z. Angew. Math. Phys. 70(122), 21 (2019)
Daus, E.S., Desvillettes, L., Dietert, H.: About the entropic structure of detailed balanced multi-species cross-diffusion equations. J. Differ. Equ. 266, 3861–3882 (2019)
Daus, E.S., Ptashnyk, M., Raithel, C.: Derivation of a fractional cross-diffusion system as the limit of a stochastic many-particle system driven by Lévy noise. Submitted for publication. arXiv:2006.00277 (2020)
Desvillettes, L., Lepoutre, T., Moussa, A., Trescases, A.: On the entropic structure of reaction-cross-diffusion systems. Commun. Partial Differ. Equ. 40, 1705–1747 (2015)
Figalli, A., Philipowski, R.: Convergence to the viscous porous medium equation and propagation of chaos. Alea 4, 185–203 (2008)
Fontbona, J., Méléard, S.: Non local Lotka–Volterra system with cross-diffusion in an heterogeneous medium. J. Math. Biol. 70, 829–854 (2015)
Golse, F.: The mean-field limit for the dynamics of large particle systems. Journées Equations aux dérivées partielles 1–47 (2003)
Ichikawa, K., Rouzimaimaiti, M., Suzuki, T.: Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete Cont. Dyn. Syst. Ser. S 5, 115–126 (2012)
Jabin, P.-E., Wang, Z.: Mean field limit for stochastic particle systems. In: Active Particles, Vol. 1, pp. 379–402. Springer, Boston (2017)
Jourdain, B., Méléard, S.: Propagation of chaos and fluctuations for a moderate model with smooth initial data. Ann. Inst. H. Poincaré Probab. Stat. 34, 727–766 (1998)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)
Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)
Lepoutre, T., Moussa, A.: Entropic structure and duality for multiple species cross-diffusion systems. Nonlinear Anal. 159, 298–315 (2017)
Lieb, E.H., Loss, M.: Analysis, 2nd edn. American Mathematical Society, Providence (2001)
Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)
Moussa, A.: From non-local to classical SKT systems: triangular case with bounded coefficients. SIAM J. Math. Anal. 52, 42–64 (2020)
Nualart, D.: The Malliavin Calculus and Related Topics. Springer, Berlin (2006)
Oelschläger, K.: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Prob. 12, 458–479 (1984)
Oelschläger, K.: On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes. Prob. Theory Relat. Fields 82, 565–586 (1989)
Oelschläger, K.: Large systems of interacting particles and the porous medium equation. J. Differ. Equ. 88, 294–346 (1990)
Seo, I.: Scaling limit of two-component interacting Brownian motions. Ann. Prob. 46, 2038–2063 (2018)
Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99 (1979)
Stevens, A.: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61, 183–212 (2000). (Erratum: 61 (2000), 2200-2200.)
Sznitman, A.S.: Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56, 311–336 (1984)
Sznitman, A.S.: Topics in propagation of chaos. In: P.L. Hennequin (ed.), École d’Été de Probabilités de Saint-Flour XIX-1989, Lecture Notes Mathematics 1464. Berlin, Springer (1991)
Zamponi, N., Jüngel, A.: Analysis of degenerate cross-diffusion population models with volume filling. Ann. Inst. H. Poincaré Anal. Nonlinear 34, 1–29 (2017)