Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T19:29:37.515Z Has data issue: false hasContentIssue false
  • English
  • Français

On maximum principles for diffusion in the presence of three diffusion paths

Published online by Cambridge University Press:  17 February 2009

Alexander I. Lee  and
James M. Hill
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This note examines maximum principles for systems of parabolic partial differential equations describing diffusion in the presence of three diffusion paths. The particular system under consideration arises from a random walk model. For a more general system constraints on the various constants are given which guarantee maximum principles. Remarkably, the physical system arising from the random walk model automatically satisfies these constraints.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 4 , April 1983 , pp. 417 - 423
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Aifantis, E. C., “A new interpretation of diffusion in high diffusivity paths—a continuum approach”, Acta Metall. 27 (1979), 683–691.CrossRefGoogle Scholar
[2]Aifantis, E. C., “Continuum basis for diffusion in regions with multiple diffusivity”, J. Appl. Phys. 50 (1979), 1334–1338.CrossRefGoogle Scholar
[3]Aifantis, E. C. and Hill, J. M., “On the theory of diffusion in media with double diffusivity I. Basic mathematical results”, Quart. J. Mech. Appl. Math. 33 (1980), 1–21.Google Scholar
[4]Barenblatt, G. I., Zheltov, lu. P. and Kochina, I. N., “Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks”, Appl. Math. Mechs. 24 (1960), 1286–1303.Google Scholar
[5]Dow, M. A., “Three-curves theorems for quasilinear inequalities”, Duke Math. J. 41 (1974), 473–481.Google Scholar
[6]Dow, M. A., “Strong maximal principles for weakly coupled systems of quasilinear parabolic inequalities”, J. Austral. Math. Soc. Ser. A 19 (1975), 103–120.CrossRefGoogle Scholar
[7]Dow, M. A. and Výbroný, R., “Maximum principles for some quasilinear second order partial differential equations”, Rend. Sem. Mat. Univ. Padova 47 (1972), 331–351.Google Scholar
[8]Hill, J. M., “A discrete random walk model for diffusion in media with double diffusivity”, J. Austral. Math. Soc. Ser. B 22 (1980), 58–74.Google Scholar
[9]Hill, J. M., “On the solution of reaction-diffusion equations”, J. Inst. Math. Appl. 27 (1981), 177–194.Google Scholar
[10]Hill, J. M. and Aifantis, E. C., “On the theory of diffusion in media with double diffusivity II. Boundary value problems”, Quart. J. Mech. Appl. Math. 33 (1980), 23–41.Google Scholar
[11]McNabb, A., “Comparison and existence theorems for multicomponent diffusion systems”, J. Math. Anal. Appl. 3 (1961), 133–144.Google Scholar
[12]Molz, F. J., “Water transport through plant tissue: the apoplasm and symplasm pathways”, J. Theoret. Biol. 59 (1976), 277–292.Google Scholar
[13]Rubinstein, L. N., “Process of conduction of heat in heterogeneous media”, Izv. Akad. Nauk SSR. Geogr. 12 (1948), 27–45.Google Scholar