Carrillo Menéndez, José and Lyaghfouri, A.
(1998)
*The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions.*
Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 26
(3).
pp. 453-505.
ISSN 0391-173X

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## Abstract

The subject of this paper is the study of a free boundary problem for a steady fluid flow through a porous medium, in which the classical Darcy law (1) −!v = ar(p(x)+xn), x = (x1, · · · , xn) 2 Rn, a > 0, is replaced by the nonlinear law (2) |−!v |m−1−!v = ar(p(x)+xn), x = (x1, · · · , xn) 2

Rn, a, m > 0, where −!v and p are, respectively, the velocity and the pressure of the fluid. This approach is particularly interesting because Darcy’s law was established on a purely experimental basis; but it is not clear why, from a physical point of view, the specific form of (2) gives a better model for the dam problem. The authors first reduce the problem to a variational inequality

involving the degenerate Laplacian operator for the hydrostatic head u(x) = p(x) + xn. Then,using a perturbation argument, they prove existence of weak solutions. The remaining portion of the paper is devoted to the study of the qualitative properties of the solutions. In particular, it is proven that the free boundary is a lower semicontinuous curve of the form xn = (x1, · · · , xn−1),and that there is a unique minimal solution.

Moreover, in the two-dimensional case the authors show that is actually continuous, and that there is a unique S3-connected solution.

Item Type: | Article |
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Subjects: | Sciences > Physics > Mathematical physics |

ID Code: | 15871 |

References: | R. A. Adams, ”Sobolev Spaces”, Academic Press, New York, 1975. H.W. Alt, A free boundary problem associated with the flow of ground water, Arch. Rat. Mech. Anal. 64 (1977), 111–126. H. W. Alt, Str¨omunger durch inhomogene por¨ose Medien mit freiem Rand, J. Reine Angew. Math. 305 (1979), 89–115. A. Alonso - J. Carrillo, A unified formulation for the boundary conditions in some convectionsdiffusion problems. Proc. European Conference on ”Elliptic and Parabolic Problems”, Pont- á-Mousson (June 1994) 325, 51–63. Pitman Research Notes in Mathematics. H. W. Alt - G. Gilardi, The behavior of the free boundary for the dam problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 9 (1981), 571–626. N. Ahmed - D. K. Sunada, Nonlinear flow in porous media, J. Hydraulics Div. Proc. Amer.Soc. Civil. Eng. 95 (1969), 1847–1857. 8. C. Baiocchi, Free boundary problems in the theory of fluid flow through porous media,Proceedings of the International Congress of Mathematicians-Vancouver (1974), 237–243. C. Baiocchi, Free boundary problems in fluid flows through porous media and variational inequalities In: ”Free Boundary Problems”, Proceedings of a seminar held in Pavia Sept-Oct 1979, Vol. 1 (Roma 1980), pp. 175–191. H. Brezis - D. Kinderlehrer - G. Stampacchia, Sur une nouvelle formulation du probléme de l’écoulement á travers une digue, C. R. Acad. Sci. Paris S´er. A 287 (1978), 711–714. H. Brezis, ”Analyse Fonctionnelle. Théorie et Applications”, Masson, 1987 J. Carrillo, ”An Evolution Free Boundary Problem: Filtrations of a Compressible Fluid in a Porous Medium”, Research Notes in Mathematics, Pitman, London, Vol. 89, 1983, pp. 97–110 J. Carrillo, On the uniqueness of the solution of the evolution dam problem, Nonlinear Anal. Vol. 22, 5 (1994), 573–607. J. Carrillo - M. Chipot, On the dam problem, J.Differential Equations 45 (1982), 234–271. J. Carrillo - M. Chipot, The dam problem with leaky boundary conditions, Appl. Math. Optim. 28 (1993), 57–85 L. A. Caffarelli - A. Friedman, The dam problem with two layers, Arch. Ration. Mech. Anal.68 (1978), 125–154. L. A. Caffarelli - A. Friedman, Asymptotic estimates for the dam problem with several layers,Indiana Univ. Math. J. 27 (1978), 551–580. L. A. Caffarelli - G. Gilardi, Monotonicity of the free boundary in the two dimensional dam problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1980), 523–537. J. Carrillo - G. Gilardi, La vitesse de propagation dans le probléme de la digue, Ann. Fac. Sci.de Toulouse Math. 11 (1990), 7–28. M. Chipot - A. Lyaghfouri, An existence theorem for an unbounded dam with leaky boundary conditions, Proc. European Conference on ”Elliptic and Parabolic Problems”, Pont-`a-Mousson 325 (June 1994), 64–73. Pitman Research Notes in Mathematics. M. Chipot - A. Lyaghfouri, The dam problem for nonlinear Darcy’s law and leaky boundary conditions, Math. Methods Appl. Sci. 20 (1997), 1045–1068. M. Chipot - A. Lyaghfouri, The dam problem for linear Darcy’s law and nonlinear leaky boundary conditions, Advances in Differential Equations, 3 (1998), 1–50. J. I. Diaz, ”Non Linear Partial Differential Equations and Free Boundaries” Vol I: Elliptic Equations, Pitman Research Notes in Mathematics, London, 1985. E. DiBenedetto, C1 local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983),827–850. E. DiBenedetto - A. Friedman, Periodic behaviour for the evolutionary dam problem and related free boundary problems, Comm. Partial Differential Equations 11 (1986), 1297–1377. Dongming Wei, An existence theorem for weak solution of a nonlinear dam problem, Appl.Anal. 34 (1989), 219–230. L. Dung, On a class of singular quasilinear elliptic equations with general structures and distributions data, Nonlinear Anal. 28 (1997), 1879–1902. A. Friedman, ”Variational Principles and Free-Boundary Problems”, Robert E. Krieger Publishing Company, Malabar, Florida, 1988. A. Friedman - Shav-Yun Huang, The inhomogeneous dam problem with discontinuous permeability,Ann. Scuola. Norm. Sup.Pisa, Cl. Sci (4) 14 (1987), 49–77. A. Friedman - A. Torelli, A free boundary problem connected with nonsteady filtration in porous media, Nonlinear Anal. 1 (1977), 503–545. G. Gilardi, A new approach to evolution free boundary problems, Comm. Partial Differential Equations 4 (1979) 1099–1123; 5 (1980), 983–984. D. Gilbarg - N. S. Trudinger, ”Elliptic Partial Differential Equations of Second Order”,Springer-Verlag, New York, 1977. R. Gariepy - W. P. Zeimer, Behavior at the boundary of solutions of quasilinear elliptic equations,Arch. Rational. Mech. Anal. 56 (1974), 372–384. R. Gariepy - W. P. Zeimer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational. Mech. Anal. 67 (1978), 25–39. J. Heinonen - T. Kilpeläinen - O. Martio, ”Nonlinear Potential Theory of Degenerate Elliptic Equations”, Oxford Science Publications, 1993 T. Kilpel¨ainen, H¨older continuity of solutions to quasilinear elliptic equations involving measures,Potential Anal. 3 (1994), 265–272. T. Kilpel¨ainen, A Rado type theorem for p-harmonic functions in the plane, Electron. J. Differential Equations 9 (1994), approx. 4pp. (electronic). J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,Indiana Univ. Math. J. 32 (1983), 849–858. J. L. Lewis, Capacitary functions in convex rings, Arch. Rational. Mech. Anal. 66 (1977),201–224. G. M. Liberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219. J.L. Lions, ”Quelques méthodes de résolution des problémes aux limites non linéaires”,Dunod/Gauthier-Villars, Paris, 1969. A. Lyaghfouri, The inhomogeneous dam problem with linear Darcy’s law and Dirichlet boundary conditions, Math. Models Methods Appl. Sci. 8 (1996), 1051–1077. J. J. Manfredi, p-harmonic functions in the plane, Proc. Amer. Math. Soc. 103, (1988), 473–478. J. M. Rakotoson, Equivalence between the growth of R B(x,r) |ru|p dy and T in the equation P[u] = T., J. Differential Equations 86 (1990), 102–122. J. F. Rodrigues, On the dam problem with leaky boundary condition, Portugal. Math. 39 (1980),399–411. G. Stampacchia, On the filtration of a fluid through a porous medium with variable cross section, Russian Math. Surveys 29 (1974), 89–102. R. Stavre - B. Vernescu, Incompressible fluid flow through a nonhomogeneous and anistropic dam, Nonlinear Anal. 9 (1985), 799–810. A. Torelli, Existence and uniqueness of the solution of a non steady free boundary problem,Boll. Un. Mat. Ital. (5), 14-B (1977), 423–466. A. Torelli, On a free boundary value problem connected with a non steady filtration phenomenon,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 33–59. P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations (7) 8 (1983), 773–817. R.E. Volker, Nonlinear flow in porous media by finite element, J. Hydraulics Div. Proc. Amer.Soc. Civil. Eng. 95 (1969), 2093–2114. |

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