Skip to main content

The porous medium equation

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1224))

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amann, H. On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 21 (1971), 125–146.

    Article  MathSciNet  MATH  Google Scholar 

  2. Angenent, S. Analyticity of the interface of the porous medium equation after the waiting time, Mathematical Institute, University of Leiden, Report No. 30, 1985.

    Google Scholar 

  3. aronson, D.G. Regularity properties of flows through porous media, SIAM J. Appl. Math., 17 (1969), 461–467.

    Article  MathSciNet  MATH  Google Scholar 

  4. Aronson, D.G. Regularity properties of flows through porous media: a counter-example, SIAM J. Appl. Math., 19 (1970), 299–307.

    Article  MathSciNet  MATH  Google Scholar 

  5. Aronson, D.G. Regularity properties of flows through porous media: the interface, Arch. Rat. Mech. Anal., 37 (1970), 1–10.

    Article  MathSciNet  MATH  Google Scholar 

  6. Aronson, D.G. Nonlinear Diffusion Problems, in Free Boundary Problems: Theory and Applications, Vol. 1 (A. Fasano and M. Primicerio, editors), Research Notes in Mathematics 78, Pitman, London, 1983, 135–149.

    Google Scholar 

  7. Aronson D.G. and Benilan, Ph. Regularite des solutions de l'equation des milieux poreux dans ℝN, C.R. Acad. Sc. Paris, 288 (1979), 103–105.

    MathSciNet  MATH  Google Scholar 

  8. Aronson, D.G. and Caffarelli, L.A. The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc., 280 (1983), 351–366.

    Article  MathSciNet  MATH  Google Scholar 

  9. Aronson, D.G. and Caffarelli, L.A. Optimal regularity for one dimensional porous medium flow, in preparation.

    Google Scholar 

  10. Aronson, D.G., Caffarelli, L.A. and Kamin, S. How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal., 14 (1983), 639–658.

    Article  MathSciNet  MATH  Google Scholar 

  11. Aronson, D.G., Caffarelli, L.A. and Vazquez, J.L. Interfaces with a corner point in one-dimensional porous medium flow, Comm. Pure Appl. Math., 38 (1985), 375–404.

    Article  MathSciNet  MATH  Google Scholar 

  12. Aronson, D.G., Crandall, M.G. and Peletier, L.A. Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. TMA, 6 (1982), 1001–1022.

    Article  MathSciNet  MATH  Google Scholar 

  13. Aronson,D.G. and Graveleau, J. In preparation.

    Google Scholar 

  14. Aronson, D.G. and Peletier, L.A. Large time behaviour of solutions of the porous medium equation in bounded domains, J. Diff. Eq., 39 (1981), 378–412.

    Article  MathSciNet  MATH  Google Scholar 

  15. Aronson, D.G. and Vazquez, J.L. The porous medium equation as a finite speed approximation to a Hamilton-Jacobi equation, Inst. for Math. and its Applications Preprint Series 143, 1985; Analyse Non Lineaire, to appear.

    Google Scholar 

  16. Aronson, D.G. and Vazquez, J.L. Eventual C-regularity and concavity for flows in one dimensional porous media, in preparation.

    Google Scholar 

  17. Barenblatt, G.I. On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Meh., 16 (1952), 67–78.

    MathSciNet  Google Scholar 

  18. Barenblatt, G.I. Similarity, Self-Similarity, and Intermediate Asymptotics, Consultants Bureau, New York, 1979.

    Book  MATH  Google Scholar 

  19. Benilan, Ph. A strong regularity Lp for solutions of the porous media equation, in Contributions to Nonlinear Partial Differential Equations (C. Bardos, A. Damlamian, J.I. Diaz and J. Hernandez editors), Research Notes in Math. 89, Pitman, London, 1983, 39–58.

    Google Scholar 

  20. Benilan, Ph., Brezis, H. and Crandall, M.G. A semilinear elliptic equation in L1(ℝN), Ann. Scuola Norm. Sup. Pisa, 2 (1975), 523–555.

    MathSciNet  MATH  Google Scholar 

  21. Benilan, Ph. and Crandall, M.G. The continuous dependence on ϕ of solutions of ut − Δϕ(u) = 0, Indiana Univ. Math. J., 30 (1981), 161–177.

    Article  MathSciNet  MATH  Google Scholar 

  22. Benilan, Ph., Crandall, M.G. and Pierre, M. Solutions of the porous medium equation in ℝN under optimal conditions on initial values, Indiana Univ. Math. J., 33, (1984) 51–87.

    Article  MathSciNet  MATH  Google Scholar 

  23. Berryman, J.G. and Holland, C.J. Stability of the separable solution for fast diffusion, Arch. Rat. Mech. Anal., 74 (1980), 279–288.

    Article  MathSciNet  MATH  Google Scholar 

  24. Bertsch, M. and Peletier, L.A. Porous medium type equations: An overview, Mathematical Institute, University of Leiden, Report No. 7, 1983.

    Google Scholar 

  25. Caffarelli, L.A. and Friedman, A. Regularity of the free boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math., 101 (1979), 1193–1281.

    Article  MathSciNet  MATH  Google Scholar 

  26. Caffarelli, L.A. and Friedman, A. Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc., 252 (1979), 99–113.

    Article  MathSciNet  MATH  Google Scholar 

  27. Caffarelli, L.A. and Friedman, A. Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J., 29 (1980), 361–391.

    Article  MathSciNet  MATH  Google Scholar 

  28. Caffarelli, L.A., Vazquez, J.L. and Wolanski, N.I. Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Inst. for Math. and its Applications. Preprint Series 191, 1985.

    Google Scholar 

  29. Caffarelli, L.A. and Wolanski, N.I. In preparation.

    Google Scholar 

  30. Chipot, M. and Sideris, T.S. An upper bound for the waiting time for non-linear degenerate parabolic equations, Trans. Amer. Math. Soc., 288 (1985), 423–427.

    Article  MathSciNet  MATH  Google Scholar 

  31. Crandall, M.G. and Lions, P.L. Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1–42.

    Article  MathSciNet  MATH  Google Scholar 

  32. Darcy, H. Les Fontaines Publiques de la Ville de Dijon, V. Dalmont, Paris, 1856, 305–311.

    Google Scholar 

  33. DiBenedetto, E. Regularity results for the porous medium equation, Ann. Mat. Pura Appl., 121 (1979), 249–262.

    Article  MathSciNet  MATH  Google Scholar 

  34. Dahlberg, B.E.J. and Kenig, C.E. Nonnegative solutions of the porous medium equation, Comm. PDE, 9 (1984), 409–437.

    Article  MathSciNet  MATH  Google Scholar 

  35. Dahlberg, B.E.J. and Kenig, C.E. Nonlinear filtration, preprint.

    Google Scholar 

  36. Friedman, A. and Kamin, S. The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math.Soc., 262 (1980), 551–563.

    MathSciNet  MATH  Google Scholar 

  37. Gilding, B.H. Holder continuity of solutions of parabolic equations, J. London Math. Soc., 13 (1976), 103–106.

    Article  MathSciNet  MATH  Google Scholar 

  38. Gilding, B.H. and Peletier, L.A. On a class of similarity solutions of the porous medium equation, J. Math. Anal. Appl., 55 (1976), 351–364: II, 57 (1977), 522–538.

    Article  MathSciNet  MATH  Google Scholar 

  39. Gilding, B.H. and Peletier, L.A. Continuity of solutions of the porous medium equation, Ann. Scuola Norm. Sup. Pisa, 8 (1981), 659–675.

    MathSciNet  MATH  Google Scholar 

  40. Graveleau, J. Personel communication, 1973.

    Google Scholar 

  41. Gurtin, M.E. and MacCamy, R.C. On the diffusion of biological populations, Math. Biosc., 33 (1977), 35–49.

    Article  MathSciNet  MATH  Google Scholar 

  42. Herrero, M.A. and M. Pierre. The Cauchy problem for ut = Δum when 0<m<1, Trans. Amer. Math. Soc., 291 (1985), 145–158.

    MathSciNet  Google Scholar 

  43. Hollig, K. and Kreiss, H.O. C-regularity for the porous medium equation, Univ. of Wisconsin-Madison, Computer Science Dept., Technical Report No. 600 1985.

    Google Scholar 

  44. Kalashnikov, A.S. The Cauchy problem in a class of growing functions for equations of unsteady filration type, Vest. Mosk. Univ. Ser. Mat. Mech., 6 (1963), 17–27.

    Google Scholar 

  45. Kamenomostskaya, S., (Kamin). The asymptotic behaviour of the solution of the filtration equation, Israel J. Math., 14 (1973), 76–78.

    Article  MathSciNet  MATH  Google Scholar 

  46. Knerr, B.F. The porous medium equation in one dimension, Trans. Amer. Math. Soc., 234 (1977), 381–415.

    Article  MathSciNet  MATH  Google Scholar 

  47. Kruzhkov, S.N. Results on the character of the regularity of solutions of parabolic equations and some of their applications, Math. Notes, 6 (1969), 517–523.

    Article  Google Scholar 

  48. Ladyzhenskaya, O.A., Solonnikov, V.A. and Ural'ceva, N.N. Linear and Quasi-linear Equations of Parabolic Type, Transl. Math. Monographs 23, Amer. Math. Soc., Providence, 1968.

    Google Scholar 

  49. Langlais, M. and Phillips, D. Stabilization of solutions of nonlinear and degenerate evolution equations, Nonlinear Anal. TMA, 9 (1985), 321–333.

    Article  MathSciNet  MATH  Google Scholar 

  50. Lions, P.L. Generalized Solutions of Hamilton-Jacobi Equations, Research Notes in Math., 69 Pitman, London, 1982.

    MATH  Google Scholar 

  51. Lions, P.L., Souganidis, P.E. and Vazquez, J.L. In preparation.

    Google Scholar 

  52. Muskat, M. The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937.

    MATH  Google Scholar 

  53. Oleinik, O.A. On some degenerate quasilinear parabolic equations, Seminari dell' Istituto Nazionale di Alta Matematica 1962–63, Oderisi, Gubbio, 1964, 355–371.

    Google Scholar 

  54. Oleinik, O.A., Kalashnikov, A.S. and Chzhou Yui-Lin. The Cauchy problem and boundary problems for equations of the type of unsteady filtration, Izv. Akad. Nauk SSSR Ser. Mat., 22 (1958), 667–704.

    MathSciNet  MATH  Google Scholar 

  55. Peletier, L.A. The porous medium equation, in Applications of Nonlinear Analysis in the Physical Sciences (H. Amann, N. Bazley and K. Kirchgassner editors), Pitman, London, 1981, 229–241.

    Google Scholar 

  56. Pierre, M. Uniqueness of the stolution of ut-Δϕ(u) = O with initial datum a measure, Nonlinear Anal. TMA, 6 (1982), 175–187.

    Article  MathSciNet  Google Scholar 

  57. Protter, M.H. and Weinberger, H.F. Maximum Principles in Differential Equations Prentice-Hall, Englewood Cliffs, 1967.

    MATH  Google Scholar 

  58. Sabinina, E.S. On the Cauchy problem for the equation of nonstationary gas filtration in several space variables, Dokl. Akad. Nauk SSSR, 136 (1961), 1034–1037.

    MathSciNet  MATH  Google Scholar 

  59. Tomoeda, K. and Mimura M. Numerical approximations to interface curves for a porous media equation, Hiroshima Math. J., 13 (1983), 273–294.

    MathSciNet  MATH  Google Scholar 

  60. Vazquez, J.L. Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc., 277 (1983), 507–527.

    Article  MathSciNet  MATH  Google Scholar 

  61. Vazquez, J.L. The interface of one-dimensional flows in porous media, Trans. Amer. Math. Soc., 285 (1984), 717–737.

    Article  MathSciNet  MATH  Google Scholar 

  62. Vazquez, J.L. Behaviour of the velocity of one-dimensional flows in porous media, Trans. Amer. Math. Soc., 286 (1984), 787–802.

    Article  MathSciNet  MATH  Google Scholar 

  63. Widder, D.V. Positive temperature on an infinite rod, Trans. Amer. Math. Soc., 55 (1944), 85–95.

    Article  MathSciNet  MATH  Google Scholar 

  64. Zeldovich, Ya.B. and Raizer, Yu.P. Physics of Shock-waves and High-temperature Hydrodynamic Phenomena Vol. II, Academic Press, New York, 1966.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Antonio Fasano Mario Primicerio

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag

About this chapter

Cite this chapter

Aronson, D.G. (1986). The porous medium equation. In: Fasano, A., Primicerio, M. (eds) Nonlinear Diffusion Problems. Lecture Notes in Mathematics, vol 1224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072687

Download citation

  • DOI: https://doi.org/10.1007/BFb0072687

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17192-8

  • Online ISBN: 978-3-540-47352-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics