On the equation of turbulent filtration in one-dimensional porous media. (English) Zbl 0613.76102

The authors consider the one-dimensional, turbulent, polytropic flow of a gas in a porous medium described by the Leibenson model (*) \(u_ t=(D(u,u_*))_ x,\quad t>0,\quad x\in {\mathbb{R}},\quad u(0,x)=u_ 0(x)\in L^ 1({\mathbb{R}}),\quad u_ 0\geq 0\) a.e. where, in the diffusion coefficient \(D(u,u_ x)=m^ pu^{p(m-1)}| u_ x|^{p-1}\) the positive constants p and m satisfy \(mp>1\). They prove the existence and uniqueness of a strong solution of (*), together with some regularity properties which imply that the free boundaries are Lipschitz-continuous, non-decreasing curves.
Reviewer: D.Polisevski


76S05 Flows in porous media; filtration; seepage
76F99 Turbulence
35K65 Degenerate parabolic equations
76R99 Diffusion and convection
Full Text: DOI


[1] Aronson, D. G., Regularity properties of flows through porous media, SIAM J. appl. Math., 17, 461-467 (1969) · Zbl 0187.03401
[2] Barenblatt, G. I., On self-similar motions of compressible fluids in porous media, Prikl. Mat. Mek., 16, 679-698 (1952), (In Russian.) · Zbl 0047.19204
[3] Benilan, Ph.; Bardos, C., A strong regularity \(L^p\) for solution of the porous media equation, Pitman Research Notes No. 86. Pitman Research Notes No. 86, Contributions to Nonlinear PDE (1983)
[4] Benilan, Ph., Sur un probleme d’evolution non monotone dans \(L^2(Ω)\), Publ. Math. Fac. Sc. Besançon (1976), No. 2
[5] Benilan, Ph.; Brezis, H.; Crandall, M. G., A semilinear equation in \(L^1(R^n)\), Annali Scu. norm. sup. Pisa, 4, 523-555 (1975) · Zbl 0314.35077
[6] Benilan, Ph.; Crandall, M. G., The continuous dependence on \(ø\) of the solutions of \(u_t = Δø (u)\), Indiana Univ. Math. J., 30, 161-177 (1981) · Zbl 0482.35012
[7] Benilan, Ph.; Crandall, M. G.; Clark, D. N.; Pecelli, C.; Sacksteder, R., Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, 23-29 (1981), supplement to Am. J. Math., Baltimore · Zbl 0556.35067
[8] Brezis, H., Opérateurs Maximaux Monotones et Semigroupes de Contraction dans les Espaces de Hilbert (1973), North Holland: North Holland Amsterdam · Zbl 0252.47055
[9] Brezis, H.; Pazy, A., Accretive sets and differential equations in Banach spaces, Israel J. Math., 8, 367-383 (1970) · Zbl 0209.45602
[10] Crandall, M. G.; Liggett, T. M., Generation of semi-groups of nonlinear transformations on general Banach spaces, Am. J. Math., 93, 265-293 (1971) · Zbl 0226.47038
[11] Evans, L. C., Application of nonlinear semigroup theory to certain partial differential equations, (Crandall, M. G., Nonlinear Evolution Equations (1978), Academic Press: Academic Press New York) · Zbl 0471.35039
[12] Friedmann, A.; Kahin, S., The asymptotic behaviour of a gas in an \(n\)-dimensional porous medium, Trans. Am. math. Soc., 262, 551-563 (1980) · Zbl 0447.76076
[13] Herrero, M. A.; Vazquez, J. L., On the propagation properties of a nonlinear degenerate parabolic equation, Communs P.D.E., 7, 1381-1402 (1982) · Zbl 0516.35041
[14] Kalashnikov, A. S., On a nonlinear equation appearing in the theory of non-stationary filtration, Trud. Semin. I.G. Petrovski, 4, 137-146 (1978), (In Russian.) · Zbl 0415.35044
[15] Kalashnikov, A. S., On the propagation of perturbations in the first boundary value problem of a doubly-nonlinear degenerate parabolic equation, Trud. Semin. I.G. Petrovski, Vol. 8, 128-134 (1982), (In Russian.) · Zbl 0494.35010
[16] Kamenomostskaya, Sh., The asymptotic behaviour of the solution of the filtration equation, Israel J. Math., 14, 76-87 (1973) · Zbl 0254.35054
[17] Knerr, B. F., The porous medium equation in one dimension, Trans. Am. math. soc., 234, 381-415 (1977) · Zbl 0365.35030
[18] Lê, C. H., Etude de la classe des opérateurs \(m\)-accrétifs de \(L^1(Ω)\) et accrétifs dans \(L^x(Ω)\), (3rd cycle thesis (1977), Univ. Paris VI)
[19] Ladyz̆enskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear an quasilinear equations of parabolic type, (Transl. Math. Monogr. (1968), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0174.15403
[20] Leibenson, L. S., General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk SSSR. Izv. Akad. Nauk SSSR, Geography and Geophysics, IX, 7-10 (1945), (In Russian.) · Zbl 0061.46108
[21] Peletier, L. A., The porous media equation, (Amann, H., Applications of Nonlinear Analysis in the Physical Sciences (1981), Pitman: Pitman London), 229-241 · Zbl 0497.76083
[22] Vazquez, J. L., Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium, Trans. Am. math. Soc., 277, 507-527 (1983) · Zbl 0528.76096
[23] Vazquez, J. L., The interface of one-dimensional flows in porous media, Trans. Am. math. Soc., 285, 717-737 (1984) · Zbl 0524.35060
[24] Vazquez, J. L., Symétrisation pour \(u_t = Δø (u)\) et applications, C.R. hebd. séanc. Acad. Sci. Paris, 295, 71-74 (1982) · Zbl 0501.35015
[25] Vazquez, J. L., Symmetrization in nonlinear parabolic equations, Portugaliae Math., 41, 339-346 (1982) · Zbl 0524.35061
[26] Widder, D. V., The Heat Equation (1975), Academic Press: Academic Press New York · Zbl 0322.35041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.