Skip to main content

Horizontal line analysis of the multidimensional porous medium equation: Existence, rate of convergence and maximum principles

Part of the Lecture Notes in Mathematics book series (LNM,volume 773)

Keywords

  • Porous Medium
  • Weak Solution
  • Regularity Property
  • Porous Medium Equation
  • Unique Weak Solution

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.99
Price excludes VAT (Canada)
  • 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. R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

    MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. _____, Regularity properties of flow through porous media: A counterexample, Ibid., 19(1970)

    Google Scholar 

  4. _____, Regularity properties of flows through porous media: The interface, Archiv Rat. Mech. Anal., 37(1970), 1–10.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. M.P. Benilan, Existence de solutions fortes pour l’équation des milieux poreux, C.R. Acad. Sc. Paris, 285A(1977), 1029–1031.

    MathSciNet  MATH  Google Scholar 

  6. H. Brezis, On some degenerate nonlinear parabolic equations, in Non-linear Functional Analysis (F.E. Browder, editor), Part I, pp. 28–38, Amer. Math. Soc., Proc. Symp. in Pure Math., 18, Providence, R.I., 1970.

    Google Scholar 

  7. H. Brezis and W. Strauss, Semilinear elliptic equations in L1, J. Math. Soc. Japan, 25(1973), 565–590.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. L.A. Caffarelli and A. Friedman, The one-phase Stefan problem and the porous medium diffusion equation: Continuity of the solution in n space dimensions, Proc. Nat. Acad. Sc., 75(1978), 2084.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North Holland and American Elsevier, Amsterdam and New York, 1976.

    MATH  Google Scholar 

  10. L.C. Evans, Applications of nonlinear semigroups to partial differential equations, Proceedings of the 1977 Madison Conference on Nonlinear Equations of Evolution, to be published by Academic Press.

    Google Scholar 

  11. J.W. Jerome, Nonlinear equations of evolution and a generalized Stefan problem, J. Differential Equations, 26(1977), 240–261.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. A.S. Kalashnikov, On the occurrence of singularities in the solutions of the equation of nonstationary filtration, Z. Vych. Mat. i. Mat. Fisiki, 7(1967), 440–444.

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. S.N. Kruzkhov, Results on the character of the regularity of solutions of parabolic equations and some of their applications, Math. Z., 6(1969), 97–108.

    Google Scholar 

  15. E.B. Leach and M. Sholander, Extended Mean Values, Amer. Math. Monthly, 85(1978), 84–90.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. J.L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Paris, 1969.

    MATH  Google Scholar 

  17. O.A. Oleinik, A.S. Kalashnikov and C. Yui-Lin, The Cauchy problem and boundary problems for equations of the type of nonstationary filtration, Izvest. Akad. Nauk SSSR, Ser. Math., 22(1958), 667–704.

    MATH  Google Scholar 

  18. M.E. Rose, Numerical methods for a porous medium equation, Report ANL-78-80, Argonne National Laboratory, Argonne, Illinois, August, 1978.

    CrossRef  Google Scholar 

  19. _____, Numerical methods for a general class of porous medium equations, Argonne National Laboratory Report, Argonne, Illinois, 1979.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. G. Stampacchia, Equations Elliptic du Second Ordre a Coefficients Discontinus, University of Montreal Press, Montreal 1966.

    MATH  Google Scholar 

  22. W.A. Strauss, On weak solutions of semi-linear hyperbolic equations, An. Acad. Brasil. Cienc., 42(1970), 645–651.

    MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1980 Springer-Verlag

About this paper

Cite this paper

Jerome, J.W. (1980). Horizontal line analysis of the multidimensional porous medium equation: Existence, rate of convergence and maximum principles. In: Watson, G.A. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094164

Download citation

  • DOI: https://doi.org/10.1007/BFb0094164

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09740-2

  • Online ISBN: 978-3-540-38562-2

  • eBook Packages: Springer Book Archive