Solutions to aggregation-diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

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Fagioli, Simone ; Radici, Emanuela (2018)
  • Subject: Mathematics - Analysis of PDEs

We investigate the existence of weak type solutions for a class of aggregation-diffusion PDEs with nonlinear mobility obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We restrict the analysis to nonnegative initial data in $L^{\infty} \cap BV$ away from vacuum and supported in a closed interval with zero-velocity boundary conditions. The main novelties of this work concern the presence of a nonlinear mobility term and the non strict monotonicity of the diffusion function. As a consequence, our result applies also to strongly degenerate diffusion equations. The conclusions are complemented with some numerical simulations.
  • References (34)
    34 references, page 1 of 4

    [1] L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zu¨rich. Birkha¨user Verlag, Basel, 2005.

    [2] A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM Journal on Applied Mathematics 63(1), 259-278 (2002)

    [3] A. Bertozzi, T. Laurent: Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys. 274 (2007), 3, 717-735;

    [4] F. Betancourt, R. Bürger and K.H. Karlsen, A strongly degenerate parabolic aggregation equation. Commun. Math. Sci. 9 (2011), no. 3, 711-742.

    [5] A. Blanchet, J. Dolbeault, and B. Perthame. Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differential Equations (2006), no. 44, p.33.

    [6] C.J. Budd, g.J. Collins, W.Z. Huang and R.D. Russell Self-similar numerical solutions of the porous medium equation using moving mesh methods Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences,357, 1754, 1047-1077, (1999). 18

    [7] M. Burger, M. Di Francesco and Y. Dolak-Struss The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion. SIAM J. Math. Anal. 38 (2006), no. 4, 1288-1315.

    [8] M. Bruna, and S. J. Chapman. Diffusion of multiple species with excluded-volume effects, Journal of Chemical Physics 137 (2012), no. 20.

    [9] S. Boi, V. Capasso, and D. Morale. Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Anal. Real World Appl. 1 (2000), no. 1, 163-176.

    [10] V. Calvez and T, Gallouët, Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up. Discrete Contin. Dyn. Syst. 36 (2016), no. 3, 1175-1208.

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