Glasnik Matematicki, Vol. 53, No. 2 (2018), 449-475.

ON THE EXISTENCE OF A SOLUTION OF A CLASS OF NON-STATIONARY FREE BOUNDARY PROBLEMS

Mahmoud Bousselsal, Abdeslem Lyaghfouri and Elmehdi Zaouche

Department of Mathematics, Labo. Part. Diff. Eq. & Hist. Maths, Ecole Normale Supérieure, 16050 Vieux-Kouba Algiers, Algeria
e-mail: bousselsal55@gmail.com

Department of Mathematics and Natural Sciences, American University of Ras Al Khaimah, Ras Al Khaimah, UAE
e-mail: abdeslem.lyaghfouri@aurak.ac.ae

Department of Mathematics, Labo. Oper. Theo. & Part. Diff. Eq.: Foundations and Applications, University of EL Oued, B. P. 789 El Oued 39000, Algeria
e-mail: elmehdi-zaouche@univ-eloued.dz

Abstract.   We consider a class of parabolic free boundary problems with heterogeneous coefficients including, from a physical point of view, the evolutionary dam problem. We establish existence of a solution for this problem. We use a regularized problem for which we prove existence of a solution by applying the Tychonoff fixed point theorem. Then we pass to the limit to get a solution of our problem. We also give a regularity result of the solutions.

2010 Mathematics Subject Classification.   76S05, 35B35

Key words and phrases.   Parabolic free boundary problems, evolution dam problem, existence, regularity

Full text (PDF) (free access)

DOI: 10.3336/gm.53.2.13

References:

  1. H. W. Alt, Strömungen durch inhomogene poröse Medien mit freiem Rand, J. Reine Angew. Math. 305 (1979), 89-115.
    MathSciNet     CrossRef

  2. C. Baiocchi, Su un problema di frontiera libera connesso a questioni di idraulica, Ann. Mat. Pura Appl. (4) 92 (1972), 107-127.
    MathSciNet     CrossRef

  3. C. Baiocchi, Free boundary problems in fluid flows through porous media and variational inequalities, in: Free boundary problems, Ist. Naz. Alta Mat. Francesco Severi, Rome 1980, 175-191.
    MathSciNet    

  4. G. Bayada and M. Chambat, Existence and uniqueness for a lubrication problem with nonregular conditions on the free boundary, Boll. Un. Mat. Ital. B (6) 3 (1984), 543-557.
    MathSciNet    

  5. A. Bermúdez, M. C. Muñiz and P. Quintela, Existence and uniqueness for a free boundary problem in aluminum electrolysis, J. Math. Anal. Appl. 191 (1995), 497-527.
    MathSciNet     CrossRef

  6. J. Carrillo, An evolution free boundary problem: filtrations of a compressible fluid in a porous medium, in: Contributions to nonlinear partial differential equations, Pitman, London, 1983, 97-110.
    MathSciNet    

  7. J. Carrillo, On the uniqueness of the solution of the evolution dam problem, Nonlinear Anal. 22 (1994), 573-607.
    MathSciNet     CrossRef

  8. J. Carrillo and M. Chipot, On the dam problem, J. Differential Equations 45 (1982), 234-271.
    MathSciNet     CrossRef

  9. J. Carrillo and M. Chipot, The dam problem with leaky boundary conditions, Appl. Math. Optim. 28 (1993), 57-85.
    MathSciNet     CrossRef

  10. J. Carrillo and G. Gilardi, La vitesse de propagation dans le problème de la digue, Ann. Fac. Sc. de Toulouse Math. (5) 11 (1990), 7-28.
    MathSciNet     CrossRef

  11. J. Carrillo and A. Lyaghfouri, The dam problem for nonlinear Darcy's law and Dirichlet boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 453-505.
    MathSciNet     CrossRef

  12. J. Carrillo and A. Lyaghfouri, A filtration problem with nonlinear Darcy's law and generalized boundary conditions, Adv. Differential Equations 5 (2000), 515-555.
    MathSciNet    

  13. S. Challal and A. Lyaghfouri, A filtration problem through a heterogeneous porous medium, Interfaces and Free Bound. 6 (2004), 55-79.
    MathSciNet     CrossRef

  14. S. Challal and A. Lyaghfouri, On a class of free boundary problems of type div(a(X) ∇ u) = -div(H(X)χ(u)), Differential and Integral Equations 19 (2006), 481-516.
    MathSciNet    

  15. M. Chicco and M. Venturino, A priori inequalities for solutions of mixed boundary-value problems in unbounded domains, Ann. Mat. Pura Appl. (4) 183 (2004), 241-259.
    MathSciNet     CrossRef

  16. M. Chipot, Variational inequalities and flow in porous media, Springer, Berlin, 1984.
    MathSciNet     CrossRef

  17. M. Chipot and M. C. Muñiz, A free boundary problem modelling the electrolysis of aluminium, Appl. Math. Optim. 47 (2003), 231-252.
    MathSciNet     CrossRef

  18. M. Chipot and A. Lyaghfouri, The dam problem for non-linear Darcy's laws and non-linear leaky boundary conditions, Math. Methods Appl. Sci. 20 (1997), 1045-1068.
    MathSciNet     CrossRef

  19. M. Chipot and A. Lyaghfouri, The dam problem with linear Darcy's law and nonlinear leaky boundary conditions, Adv. Differential Equations 3 (1998), 1-50.
    MathSciNet    

  20. E. DiBenedetto, A. Friedman, Periodic behaviour for the evolutionary dam problem and related free boundary problems, Comm. Partial Differential Equations 11 (1986), 1297-1377.
    MathSciNet     CrossRef

  21. G. Gilardi, A new approach to evolution free boundary problems, Comm. Partial Differential Equations 4 (1979), 1099-1122; 5 (1980), 983-984.
    MathSciNet     CrossRef
    MathSciNet    

  22. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, New York, 1983.
    MathSciNet     CrossRef

  23. A. Lyaghfouri, The inhomogeneous dam problem with linear Darcy's law and Dirichlet boundary conditions, Math. Models Methods Appl. Sci. 6 (1996), 1051-1077.
    MathSciNet     CrossRef

  24. A. Lyaghfouri, The evolution dam problem for nonlinear Darcy's law and nonlinear leaky boundary conditions, Ricerche di Matematica (2) XLVII (1998), 297-357.
    MathSciNet    

  25. A. Lyaghfouri, A unified formulation for the dam problem, Riv. Mat. Univ. Parma (6) 1 (1998), 113-148.
    MathSciNet    

  26. A. Lyaghfouri, The evolution dam problem for nonlinear Darcy's law and Dirichlet boundary conditions, Portugal. Math. 56 (1999), 1-37.
    MathSciNet    

  27. A. Lyaghfouri, A regularity result for a heterogeneous evolution dam problem, Z. Anal. Anwendungen 24 (2005), 149-166.
    MathSciNet     CrossRef

  28. A. Lyaghfouri and E. Zaouche, Uniqueness of solution of the unsteady filtration problem in heterogeneous porous media, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 112 (2018), 89-102.
    MathSciNet     CrossRef

  29. J. F. Rodrigues, On the dam problem with leaky boundary condition, Portugal. Math. 39 (1980), 399-411.
    MathSciNet    

  30. D. R. Smart, Fixed point theorems, Cambridge University Press, Cambridge, 1974.
    MathSciNet    

  31. A. Torelli, Existence and uniqueness of the solution of a non steady free boundary problem, Boll. Un. Mat. Ital. B (5) 14 (1977), 423-466.
    MathSciNet    

  32. E. Zaouche, Uniqueness of solution in a rectangular domain of an evolution dam problem with heterogeneous coefficients, Electron. J. Differ. Equations (169) 2018 (2018), 1-17.
Glasnik Matematicki Home Page