Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium
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- by Juan Luis Vázquez
- Trans. Amer. Math. Soc. 277 (1983), 507-527
- DOI: https://doi.org/10.1090/S0002-9947-1983-0694373-7
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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
- D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461–467. MR 247303, DOI 10.1137/0117045
- D. G. Aronson, Regularity properties of flows through porous media: The interface, Arch. Rational Mech. Anal. 37 (1970), 1–10. MR 255996, DOI 10.1007/BF00249496
- Donald G. Aronson and Philippe Bénilan, Régularité des solutions de l’équation des milieux poreux dans $\textbf {R}^{N}$, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 2, A103–A105 (French, with English summary). MR 524760 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.
- D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains, J. Differential Equations 39 (1981), no. 3, 378–412. MR 612594, DOI 10.1016/0022-0396(81)90065-6
- G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 67–78 (Russian). MR 0046217 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. Ph. Bénilan, Equations d’évolution dans un espace de Banach quelconque et applications, Thesis, Univ. Orsay, 1972. 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.
- Philippe Bénilan and Michael G. Crandall, The continuous dependence on $\varphi$ of solutions of $u_{t}-\Delta \varphi (u)=0$, Indiana Univ. Math. J. 30 (1981), no. 2, 161–177. MR 604277, DOI 10.1512/iumj.1981.30.30014
- Luis A. Caffarelli and Avner Friedman, Regularity of the free boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math. 101 (1979), no. 6, 1193–1218. MR 548877, DOI 10.2307/2374136
- Michael G. Crandall, An introduction to evolution governed by accretive operators, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974) Academic Press, New York, 1976, pp. 131–165. MR 0636953
- Avner Friedman and Shoshana Kamin, The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980), no. 2, 551–563. MR 586735, DOI 10.1090/S0002-9947-1980-0586735-0
- A. S. Kalašnikov, Formation of singularities in solutions of the equation of nonstationary filtration, Ž. Vyčisl. Mat i Mat. Fiz. 7 (1967), 440–444 (Russian). MR 211058
- S. Kamenomostskaya, The asymptotic behavior of the solution of the filtration equation, Israel J. Math. 14 (1973), 76–87. MR 315292, DOI 10.1007/BF02761536
- Barry F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), no. 2, 381–415. MR 492856, DOI 10.1090/S0002-9947-1977-0492856-3
- A. A. Lacey, J. R. Ockendon, and A. B. Tayler, “Waiting-time” solutions of a nonlinear diffusion equation, SIAM J. Appl. Math. 42 (1982), no. 6, 1252–1264. MR 678215, DOI 10.1137/0142087
- O. A. Oleĭnik, A. S. Kalašinkov, and Yuĭ-Lin′Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 667–704 (Russian). MR 0099834
- L. A. Peletier, The porous media equation, Applications of nonlinear analysis in the physical sciences (Bielefeld, 1979), Surveys Reference Works Math., vol. 6, Pitman, Boston, Mass.-London, 1981, pp. 229–241. MR 659697 L. Veron, Coercivité et propriétés régularisantes des semi-groupes non linéaires dans les espaces de Banach, Publ. Fac. Sci. Besançon 3 (1977). 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.
Bibliographic Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 277 (1983), 507-527
- MSC: Primary 35B40; Secondary 35K55, 76S05
- DOI: https://doi.org/10.1090/S0002-9947-1983-0694373-7
- MathSciNet review: 694373