Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium

Author: Juan Luis Vázquez
Journal: Trans. Amer. Math. Soc. 277 (1983), 507-527
MSC: Primary 35B40; Secondary 35K55, 76S05
MathSciNet review: 694373
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The one-dimensional porous media equation $ {u_t} = {({u^m})_{xx}}$, $ m > 1$, is considered for $ x \in R$, $ t > 0$ with initial conditions $ u(x,0) = {u_0}(x)$ integrable, nonnegative and with compact support. We study the behaviour of the solutions as $ t \to \infty $ proving that the expressions for the density, pressure, local velocity and interfaces converge to those of a model solution. In particular the first term in the asymptotic development of the free-boundary is obtained.

References [Enhancements On Off] (What's this?)

  • [1] D. G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461-467. MR 0247303 (40:571)
  • [2] -, Regularity properties of flows through porous media: The interface, Arch. Rational Mech. Anal. 37 (1970), 1-10. MR 0255996 (41:656)
  • [3] D. G. Aronson and Ph. Bénilan, Régularité de l'équation des milieux poreux dans $ {{\mathbf{R}}^N}$, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), 103-105. MR 524760 (82i:35090)
  • [4] D. G. Aronson, L. A. Caffarelli and S. Kamin, How an initially stationary interface begins to move in porous medium flow, Univ. Minnesota Math. Report No. 81-113, 1981.
  • [5] D. G. Aronson and L. A. Peletier, Large-time behaviour of solutions of the porous media equation in bounded domains, J. Differential Equations 39 (1981), 378-412. MR 612594 (82g:35047)
  • [6] G. I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Akad. Nauk SSSR Prikl. Mat Meh. 16 (1952), 67-78. (Russian) MR 0046217 (13:700a)
  • [7] G. I. Barenblatt and Ya. B. Zeldovich, The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations, Soviet Physics Dokl. 3 (1958), 44-47.
  • [8] Ph. Bénilan, Equations d'évolution dans un espace de Banach quelconque et applications, Thesis, Univ. Orsay, 1972.
  • [9] Ph. Bénilan, H. Brézis and M. G. Crandall, A semilinear elliptic equation in $ {L^1}({{\mathbf{R}}^N})$, Ann. Scuola Norm. Sup. Pisa 4 (1975), 523-555.
  • [10] Ph. Bénilan and M. G. Crandall, The continuous dependence on $ \phi$ of the solutions of $ {u_t} - \Delta \phi (u) = 0$, Indiana Univ. Math. J. 30 (1981), 161-177. MR 604277 (83d:35071)
  • [11] L. A. Caffarelli and A. Friedman, Regularity of the free-boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math. 101 (1979), 1193-1218. MR 548877 (80k:76072)
  • [12] M. G. Crandall, An introduction to evolution governed by accretive operators, Dynamical Systems--An International Symposium (L. Cesari et al., Editors), Academic Press, New York, 1976, pp. 131-165. MR 0636953 (58:30550)
  • [13] A. Friedman and S. Kamin, The asymptotic behaviour of a gas in an $ n$-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980), 551-563. MR 586735 (81j:35054)
  • [14] A. S. Kalashnikov, Formation of singularities in solutions of the equation of nonstationary filtration, Ž. Vyčisl. Mat. i Mat. Fiz. $ {\mathbf{7}}$ (1967), 440-444. MR 0211058 (35:1940)
  • [15] Sh. Kamenomostskaya (Kamin), The asymptotic behaviour of the solution of the filtration equation, Israel J. Math. 14 (1973), 76-78. MR 0315292 (47:3841)
  • [16] B. F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), 381-415. MR 0492856 (58:11917)
  • [17] A. A. Lacey, J. R. Ockendon and A. B. Tayler, 'Waiting-time' solutions of a nonlinear diffusion equation, Preprint (1981). MR 678215 (84f:80008)
  • [18] O. A. Oleinik, A. S. Kalashinikov and Czhou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 667-704. (Russian) MR 0099834 (20:6271)
  • [19] L. A. Peletier, The porous media equation, Applications of Nonlinear Analysis in the Physical Sciences (H. Amann et al., Editors), Pitman, London, 1981, pp. 229-241. MR 659697 (83k:76076)
  • [20] L. Veron, Coercivité et propriétés régularisantes des semi-groupes non linéaires dans les espaces de Banach, Publ. Fac. Sci. Besançon 3 (1977).
  • [21] Ya. B. Zeldovich and Yu. P. Raizer, Physics of shock-waves and high-temperature hydrodynamic phenomena. Vol. II, Academic Press, New York, 1966, p. 681.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35B40, 35K55, 76S05

Retrieve articles in all journals with MSC: 35B40, 35K55, 76S05

Additional Information

Keywords: Flows in porous media, asymptotic behaviour, free boundaries, shiftingcomparison
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society