In this paper, a diffusion-aggregation equation with delta potential is introduced. Based on the global existence and uniform estimates of solutions to the diffusion-aggregation equation, we also provide the rigorous derivation from a stochastic particle system while introducing an intermediate particle system with smooth interaction potential. The theoretical results are compared to numerical simulations relying on suitable discretization schemes for the microscopic and macroscopic level. In particular, the regime switch where the analytic theory fails is numerically analyzed very carefully and allows for a better understanding of the equation.
Mots-clés : Interacting particle system, stochastic processes, mean-field equations, hydrodynamic limit, numerical simulations
@article{M2AN_2018__52_2_567_0,
author = {Chen, Li and G\"ottlich, Simone and Knapp, Stephan},
title = {Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {567--593},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2018028},
zbl = {1404.35434},
mrnumber = {3834436},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2018028/}
}
TY - JOUR AU - Chen, Li AU - Göttlich, Simone AU - Knapp, Stephan TI - Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 567 EP - 593 VL - 52 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2018028/ DO - 10.1051/m2an/2018028 LA - en ID - M2AN_2018__52_2_567_0 ER -
%0 Journal Article %A Chen, Li %A Göttlich, Simone %A Knapp, Stephan %T Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 567-593 %V 52 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2018028/ %R 10.1051/m2an/2018028 %G en %F M2AN_2018__52_2_567_0
Chen, Li; Göttlich, Simone; Knapp, Stephan. Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 567-593. doi : 10.1051/m2an/2018028. http://archive.numdam.org/articles/10.1051/m2an/2018028/
[1] and , Stochastic Simulation: Algorithms and Analysis. Vol. 57 of Stochastic Modelling and Applied Probability. Springer, New York (2007). | DOI | MR | Zbl
[2] , Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models. Commun. Math. Sci. 9 (2011) 1143–1161. | DOI | MR | Zbl
[3] , and , A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér. 31 (1997) 615–641. | DOI | Numdam | MR | Zbl
[4] and , Finite-time blow-up of L∞-weak solutions of an aggregation equation. Commun. Math. Sci. 8 (2010) 45–65. | DOI | MR | Zbl
[5] and , A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34 (2012) B559–B583. | DOI | MR | Zbl
[6] and , An integro-differential equation arising as a limit of individual cell-based models. J. Differ. Equ. 222 (2006) 341–380. | DOI | MR | Zbl
[7] , and , On an aggregation model with long and short range interactions. Nonlinear Anal. Real World Appl. 8 (2007) 939–958. | DOI | MR | Zbl
[8] , , and , Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension. ESAIM: M2AN 42 (2008) 535–563. | DOI | Numdam | MR | Zbl
[9] , , , and , Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156 (2011) 229–271. | DOI | MR | Zbl
[10] , and , Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamer. 19 (2003) 971–1018. | DOI | MR | Zbl
[11] , and , A note on aubin-lions-dubinskiĭ lemmas. Acta Appl. Math. 133 (2014) 33–43. | DOI | MR | Zbl
[12] and , A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111 (2008) 169–205. | DOI | MR | Zbl
[13] , , and , On a chemotaxis model with saturated chemotactic flux. Kinet. Relat. Model. 5 (2012) 51–95. | DOI | MR | Zbl
[14] and , Optimal critical mass in the two-dimensional Keller-Segel model in ℝ2. C. R. Math. Acad. Sci. Paris 339 (2004) 611–616. | DOI | MR | Zbl
[15] , The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions. Commun. Math. Phys. 304 (2011) 649–664. | DOI | MR | Zbl
[16] and , Microscopic derivation of the keller-segel equation in the sub-critical regime, Preprint ArXiv: (2017). | arXiv
[17] and Propagation of chaos for a subcritical Keller-Segel model. Ann. Inst. Henri Poincaré Probab. Statist. 51 (2015) 965–992. | DOI | Numdam | MR | Zbl
[18] and , Error estimate of a random particle blob method for the Keller-Segel equation. Math. Comput. 86 (2017) 2719–2744. | DOI | MR | Zbl
[19] and , On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329 (1992) 819–824 | DOI | MR | Zbl
[20] and , Chemotaxis: from kinetic equations to aggregate dynamics. NoDEA Nonlin. Differ. Equ. Appl. 20 (2013) 101–127. | DOI | MR | Zbl
[21] and , Numerical methods for one-dimensional aggregation equations. SIAM J. Numer. Anal. 53 (2015) 895–916. | DOI | MR | Zbl
[22] and , Propagation of chaos and fluctuations for a moderate model with smooth initial data. Ann. Inst. Henri Poincaré Probab. Statist. 34 (1998) 727–766. | DOI | Numdam | MR | Zbl
[23] and , Numerical Solution of Stochastic Differential Equations. Springer, Berlin, Heidelberg (1992). | DOI | MR | Zbl
[24] , Local and global existence for an aggregation equation. Commun. Part. Differ. Equ. 32 (2007) 1941–1964. | DOI | MR | Zbl
[25] , Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). | DOI | MR | Zbl
[26] and , A random particle blob method for the Keller-Segel equation and convergence analysis. Math. Comput. 86 (2017) 725–745. | DOI | MR | Zbl
[27] , and , High order finite difference WENO schemes for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 33 (2011) 939–965. | DOI | MR | Zbl
[28] , Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Vol. 53 of Applied Mathematical Sciences. Springer-Verlag, New York (1984). | DOI | MR | Zbl
[29] , Large systems of interacting particles and the porous medium equation. J. Differ. Equ. 88 (1990) 294–346. | DOI | MR | Zbl
[30] , A partial upwind difference scheme for nonlinear parabolic equations. J. Comput. Appl. Math. 26 (1989) 219–233. | DOI | MR | Zbl
[31] , Interacting diffusions approximating the porous medium equation and propagation of chaos. Stoch. Process. Appl. 117 (2007) 526–538. | DOI | MR | Zbl
[32] , Compact sets in the space Lp (O, T ; B). Ann. Math. Pura Appl. 146 (1986) 65–96. | DOI | MR | Zbl
[33] , Topics in propagation of chaos, in École d’Été de Probabilités de Saint-Flour XIX – 1989. Vol. 1464 of Lect. Notes Math. Springer, Berlin (1991) 165–251. | MR | Zbl
[34] , and , A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68 (2006) 1601–1623. | DOI | MR | Zbl
[35] , One-dimensional kinetic models of granular flows. ESAIM: M2AN 34 (2000) 1277–1291. | DOI | Numdam | MR | Zbl
[36] , and , On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations. J. Sci. Comput. 48 (2010) 339–349. | DOI | MR | Zbl
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