Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T08:51:04.810Z Has data issue: false hasContentIssue false
  • English
  • Français

A meeting point of entropy and bifurcations in cross-diffusion herding†

Published online by Cambridge University Press:  15 August 2016

ANSGAR JÜNGEL ,
CHRISTIAN KUEHN  and
LARA TRUSSARDI
ANSGAR JÜNGEL
Affiliation:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria emails: [email protected], [email protected]
CHRISTIAN KUEHN
Affiliation:
Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Germany email: [email protected]
LARA TRUSSARDI
Affiliation:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria emails: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A cross-diffusion system modelling the information herding of individuals is analysed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

Keywords

Information herdingentropy methodlarge-time dynamics of solutionsnumerical continuationbifurcation35K5735K2035B4035Q91
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 2 , April 2017 , pp. 317 - 356
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

AJ and LT acknowledge partial support from the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617, the Austrian Science Fund (FWF), grants P22108, P24304, W1245, and the Austrian-French Program of the Austrian Exchange Service (ÖAD). CK acknowledges partial support by an APART fellowship of the Austrian Academy of Sciences (ÖAW) and by a Marie-Curie International Reintegration Grant by the EU/REA (IRG 271086).

References

[1] Arnold, A., Abdallah, N. B. & Negulescu, C. (1996) Liapunov functionals and large-time-asymptotics of mean-field nonlinear Fokker-Planck equations. Transp. Theory Stat. Phys. 25 (7), 733–751.Google Scholar
[2] Achleitner, F. & Kuehn, C. (2015) On bounded positive stationary solutions for a nonlocal Fisher-KPP equation. Nonl. Anal. A: Theor. Meth. Appl. 112, 15–29.Google Scholar
[3] Amann, H. (1989) Dynamic theory of quasilinear parabolic systems. III. Global existence. Math. Z. 202, 219–250.Google Scholar
[4] Arnold, A., Markowich, P., Toscani, G. & Unterreiter, A. (2001) On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Equ. 26 (1–2), 43–100.CrossRefGoogle Scholar
[5] Agranovich, M. S. & Vishik, M. I. (1964) Elliptic problems with a parameter and parabolic problems of general type. Russ. Math. Surv. 19 (3), 53–157.Google Scholar
[6] Bakry, D., Gentil, I. & Ledoux, M. (2014) Analysis and Geometry of Markov Diffusion Operators, Springer.Google Scholar
[7] Burger, M., Markowich, P. & Pietschmann, J.-F. (2011) Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. Kinet. Relat. Mod. 4, 1025–1047.CrossRefGoogle Scholar
[8] Carrillo, J. A., Jüngel, A., Markowich, P. A., Toscani, G. & Unterreiter, A. (2001) Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatshefte für Math. 133 (1), 1–82.Google Scholar
[9] Chertock, A., Kurganov, A., Wang, X. & Wu, Y. (2012) On a chemotaxis model with saturated chemotactic flux. Kinet. Relat. Mod. 5, 51–95.Google Scholar
[10] Crandall, M. G. & Rabinowitz, P. H. (1971) Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (2), 321–340.Google Scholar
[11] Crandall, M. G. & Rabinowitz, P. H. (1973) Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52, 161–180.Google Scholar
[12] Doedel, E. J., Champneys, A., Dercole, F., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Paffenroth, R., Sandstede, B., Wang, X. & Zhang, C. (2007) Auto 2007p: Continuation and bifurcation software for ordinary differential equations (with homcont). URL: http://cmvl.cs.concordia.ca/auto, accessed July 1st, 2016.Google Scholar
[13] Desvillettes, L. & Fellner, K. (2006) Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations. J. Math. Anal. Appl. 319 (1), 157–176.Google Scholar
[14] Desvillettes, L. & Fellner, K. (2007) Entropy methods for reaction-diffusion systems. Discrete Cont. Dyn. Sys. (suppl.) 304–312.Google Scholar
[15] Dreher, M. & Jüngel, A. (2012) Compact families of piecewise constant functions in L p (0,T;B). Nonlin. Anal. 75, 3072–3077.Google Scholar
[16] Delitala, M. & Lorenzo, T. (2014) A mathematical model for value estimation with public information and herding. Kinet. Relat. Mod. 7, 29–44.Google Scholar
[17] Dankowicz, H. & Schilder, F. (2013) Recipes for Continuation. SIAM.Google Scholar
[18] Evans, L. C. (2002) Partial Differential Equations, AMS.Google Scholar
[19] Fife, P. C. (1973) Semilinear elliptic boundary value problems with small parameters. Arch. Ration. Mech. Anal. 52 (3), 205–232.Google Scholar
[20] Gabriel, P. (2012) Long-time asymptotics for nonlinear growth-fragmentation equations. Commun. Math. Sci. 10, 787–820.CrossRefGoogle Scholar
[21] Govaerts, W. F. (1987) Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, PA.Google Scholar
[22] Galiano, G. & Selgas, V. (2014) On a cross-diffusion segregation problem arising from a model of interacting particles. Nonlin. Anal.: Real World Appl. 18, 34–49.Google Scholar
[23] Henderson, M. E. (2002) Multiple parameter continuation: Computing implicitly defined k-manifolds. Int. J. Bif. Chaos 12 (3), 451–476.CrossRefGoogle Scholar
[24] Hittmeir, S. & Jüngel, A. (2011) Cross diffusion preventing blow up in the two-dimensional Keller-Segel model. SIAM J. Math. Anal. 43, 997–1022.Google Scholar
[25] Horstmann, D. (2011) Generalizing the keller-segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species. J. Nonlin. Sci. 21, 231–270.Google Scholar
[26] Hillen, T. & Painter, K. (2002) Volume filling and quorum sensing in models for chemosensitive movement. Canad. Appl. Math. Quart. 10, 501–543.Google Scholar
[27] Januauskas, A. (1998) Classification of second-order partial differential equation systems elliptic in the petrovskii sense. Lithuanian Math. J. 38, 59–63.Google Scholar
[28] Jüngel, A. (2015) The boundedness-by-emtropy method for cross-diffusion systems. Nonlinearity 28, 1963–2001.Google Scholar
[29] Jiang, J. & Zhang, Y. (2009) On convergence to equilibria for a chemotaxis model with volume-filling effect. Asympt. Anal. 65, 79–102.Google Scholar
[30] Keller, H. (1977) Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz, P. (editor), Applications of Bifurcation Theory, Academic Press, pp. 359–384.Google Scholar
[31] Kielhoefer, H. (2004) Bifurcation Theory: An Introduction with Applications to PDEs, Springer.Google Scholar
[32] Krauskopf, B., Osinga, H. M. & Galán-Vique, J. (editors) (2007) Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Springer.Google Scholar
[33] Keller, E. & Segel, S. (1970) Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415.Google Scholar
[34] Kuehn, C. (2015). Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs. Applied Mathematics and Computation, 266, 656–674.Google Scholar
[35] Lions, P.-L. (2015) Some new classes of nonlinear Kolmogorov equations. Talk at the 16th Pauli Colloquium, Wolfgang-Pauli Institute.Google Scholar
[36] Liero, M. & Mielke, A. (2013) Gradient structures and geodesic convexity for reaction-diffusion systems. Phil. Trans. Roy. Soc. A 371, 20120346.Google Scholar
[37] Lambda, H. & Seaman, T. (2008) Market statistics of a psychology-based heterogeneous agent model. Intern. J. Theor. Appl. Finance 11, 717–737.Google Scholar
[38] Ni, W.-M. (1998) Diffusion, cross-diffusion and their spike-layer steady states. Not. Amer. Math. Soc. 45 (1), 9–18.Google Scholar
[39] Shi, J. & Wang, X. (2009) On the global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246, 2788–2812.Google Scholar
[40] Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer.Google Scholar
[41] Uhlenbeck, K. (1972) Eigenfunctions of Laplace operators. Bull. Amer. Math. Soc. 78, 1073–1076.Google Scholar
[42] Uecker, H., Wetzel, D. & Rademacher, J. D. M. (2014) pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems. Num. Math.: Th. Meth. Appl. 7, 58–106.Google Scholar
[43] Wolansky, G. (2002) Multi-components chemotactic system in the absence of conflicts. Europ. J. Appl. Math. 13, 641–661.Google Scholar
[44] Wrzosek, D. (2004) Global attractor for a chemotaxis model with prevention of overcrowding. Nonlin. Anal. 59, 1293–1310.Google Scholar
[45] Wang, X. & Xu, Q. (2013) Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem. J. Math. Biol. 66 (6), 1241–1266.Google Scholar
[46] Zinsl, J., & Matthes, D. (2015). Transport distances and geodesic convexity for systems of degenerate diffusion equations. Calculus of Variations and Partial Differential Equations, 54 (4), 3397–3438.Google Scholar