Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-19T11:08:22.694Z Has data issue: false hasContentIssue false
  • English
  • Français

Solutions of the porous medium equation with degenerate interfaces

Published online by Cambridge University Press:  07 December 2012

J. IAIA  and
S. BETELU
J. IAIA
Affiliation:
University of North Texas, Denton, TX 76203, US emails: iaia@unt.edu, betelu@unt.edu
S. BETELU
Affiliation:
University of North Texas, Denton, TX 76203, US emails: iaia@unt.edu, betelu@unt.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the existence of a one-parameter family of solutions of the porous medium equation in which the interface is a half line whose end point advances at a constant speed. Then we prove the stability of the solutions under a suitable class of perturbations. We discuss the relevance of these solutions to gravity-driven flows of thin films, and show that some solutions develop a very thin triangular plateau in the direction of propagation and that the angle of the plateau and its thickness are decreasing functions of the speed.

Keywords

Porous mediaGravity currentsThin filmsSingularities
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 3 , June 2013 , pp. 315 - 341
Copyright
Copyright © Cambridge University Press 2012

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.)

References

[1]Angenent, S. B., Aronson, D. G., Betelú, S., Diez, J., Gratton, J., Gratton, R., Marino, B. & Thomas, L. P. (1998) Non-circular focusing flow in viscous gravity currents. Phys. Rev. E 58, 6182–6187.Google Scholar
[2]Aronson, D. G. (1986) The porous medium equation. In: Fasano, A. & Primicerio, M. (editors), Some Problems in Nonlinear Diffusion, Lecture Notes in Math. No. 1224, Springer-Verlag, New York, pp. 1–46.Google Scholar
[3]Aronson, D. G., Caffarelli, L. A. & Kamin, S. (1983) How an initially stationary interface begins to move in porous medium flow. SIAM J. Math. Anal. 14, 639–658.Google Scholar
[4]Barenblatt, G. I. (1952) On some unsteady motions of fluids and gases in a porous medium. Prikl. Mat. Mekh. 16, 67.Google Scholar
[5]Barenblatt, G. I. (1996) Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
[6]Betelu, S. (2000) A two-dimensional corner solution for a nonlinear diffusion equation. Appl. Math. Lett. 13, 119–123.Google Scholar
[7]Betelu, S. & Diez, J. (1999) A two-dimensional similarity solution for capillary-driven flows. Physica D 126, 136.Google Scholar
[8]Betelú, S., Diez, J., Gratton, R., Marino, B. & Thomas, L. (1996) Waiting time solutions of a non-linear diffusion equation: Experimental study of a creeping flow near a waiting front. Phys. Rev. E. 54, 2628–2636.Google Scholar
[9]Birkoff, G. & Rota, G. C. (1962) Ordinary Differential Equations, Ginn and Company, California.Google Scholar
[10]Huppert, H. (1982) The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.Google Scholar
[11]Vazquez, J. L. (2007) The Porous Medium Equation, Oxford Mathematical Monographs, Oxford University Press, Oxford, UK.Google Scholar