Abstract
The nonlinear diffusion equation in bounded geometry with time-independent boundary conditions has a uniquely determined stationary solution. We show that this solution is dynamically stable in the sense of Liapunov. Any initial distribution tends to the stationary one as time goes on. It is shown that the application of the Glansdorff-Prigogine stability criterion requires a more elaborate analysis. We develop a variational procedure which has application in a wide range of nonlinear transport problems.
Similar content being viewed by others
References
Carslaw, H.S., Jaeger, J.C.: Conduction of heat in solids (2nd ed.) Oxford: Clarendon Press 1973
Crank, J.: The mathematics of diffusion (2nd ed.) Oxford: Clarendon Press 1975
Jost, W.: Diffusion in solids, liquids, gases. New York: Academic Press 1970
Ames, W.F.: Nonlinear partial differential equations in engineering. New York: Academic Press 1965
Weiss, P.: Non-equilibrium thermodynamics, variational techniques and stability (eds. Donnelly, R.J., Herman, R., Prigogine, I.) p. 295. Chicago: University of Chicago Press 1966
Glansdorff, P., Prigogine, I.: Thermodynamic theory of structure, stability and fluctuations. London: Wiley 1971
Kirchhoff, G.: Vorlesungen über die Theorie der Wärme (1894)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kern, W., Felderhof, B.U. Stability of nonlinear diffusion. Z Physik B 28, 129–134 (1977). https://doi.org/10.1007/BF01325451
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01325451