Abstract
A cross-diffusion system for two components with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker–Planck equation for the probability density associated with a multi-dimensional Itō process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser, H.J., Triebel, H. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis, p. 9126. Teubner, Stuttgart (1993)
Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations. Commun. Part. Differ. Eqs. 26, 43–100 (2001)
Chen, X., Daus, E., Jüngel, A.: Global existence analysis of cross-diffusion population systems for multiple species. Submitted for publication (2016). arXiv:1608.03696
Clark, D.: Short proof of a discrete Gronwall inequality. Discrete Appl. Math. 16, 279–281 (1987)
Desvillettes, L., Lepoutre, T., Moussa, A., Trescases, A.: On the entropic structure of reaction-cross diffusion systems. Commun. Part. Differ. Eqs. 40, 1705–1747 (2015)
Dreher, M., Jüngel, A.: Compact families of piecewise constant functions in \(L^p(0, T;B)\). Nonlinear Anal. 75, 3072–3077 (2012)
Lions, P.-L.: Some new classes of nonlinear Kolmogorov equations. Talk at the 16th Pauli Colloquium, Wolfgang-Pauli Institute, Vienna, November 18 (2015)
Jüngel, A.: The boundedness-by-entropy method for cross-diffusion systems. Nonlinearity 28, 1963–2001 (2015)
Menz, S., Latorre, J., Schütte, C., Huisinga, W.: Hybrid stochastic-deterministic solution of the chemical master equation. SIAM Multiscale Model. Simul. 10, 1232–1262 (2012)
Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (2003)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)
Villani, C.: Optimal Transport. Old and New. Springer, Berlin (2009)
Zamponi, N., Jüngel, A.: Global existence analysis for degenerate energy-transport models for semiconductors. J. Differ. Eqs. 258, 2339–2363 (2015)
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF).
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors acknowledge partial support from the Austrian Science Fund (FWF), Grants P22108, P24304, and W1245, and from the Austrian-French Project Amadeé of the Austrian Exchange Service (ÖAD), Grant FR 04/2016.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Jüngel, A., Zamponi, N. A cross-diffusion system derived from a Fokker–Planck equation with partial averaging. Z. Angew. Math. Phys. 68, 28 (2017). https://doi.org/10.1007/s00033-017-0772-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-017-0772-1