Glasnik Matematicki, Vol. 55, No. 1 (2020), 93-99.

UNIQUENESS OF SOLUTION OF A HETEROGENEOUS EVOLUTION DAM PROBLEM ASSOCIATED WITH A COMPRESSIBLE FLUID FLOW THROUGH A RECTANGULAR POROUS MEDIUM

Elmehdi Zaouche

Department of Mathematics, University of EL Oued, B. P. 789 El Oued 39000, Labo. Part. Diff. Eq. & Hist. Maths, Ecole Normale Supérieure, 16050 Vieux-Kouba Algiers, Algeria
e-mail: elmehdi-zaouche@univ-eloued.dz

Abstract.   This paper is concerned with the uniqueness of a weak solution of an evolution dam problem arising in a compressible fluid flow through a two-dimensional, rectangular, and heterogeneous porous medium. Our problem is associated with the equation a(x1)(ux2+χ)x2-(u+χ)t=0. The technique we use is based on a transformation of the weak form of this equation into a similar one that enables us to argue as in [12].

2010 Mathematics Subject Classification.   35A02, 35B35, 76S05

Key words and phrases.   Heterogeneous evolution dam problem, compressible fluid flow, rectangular porous medium, uniqueness

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.55.1.08

References:

  1. S. J. Alvarez and R. Oujja, On the uniqueness of the solution of an evolution free boundary problem in theory of lubrication, Nonlinear Anal. 54 (2003), 845-872.
    MathSciNet     CrossRef

  2. M. Bousselsal, A. Lyaghfouri and E. Zaouche, On the existence of a solution of a class of non-stationary free boundary problems, Glas. Mat. Ser. III 53(73) (2018), 449-475.
    MathSciNet     CrossRef

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

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

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

  6. G. Gilardi, A new approach to evolution free boundary problems, Comm. Partial Differential Equations 4 (1979), 1099-1123; 5 (1980), 983-984.
    MathSciNet    
    MathSciNet     CrossRef

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

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

  9. A. Lyaghfouri and E. Zaouche, Lp-continuity of solutions to parabolic free boundary problems, Electron. J. Differ. Equations 2015, No. 184, 9 pp.
    MathSciNet    

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

  11. 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    

  12. E. Zaouche, Uniqueness of solution in a rectangular domain of an evolution dam problem with heterogeneous coefficients, Electron. J. Differ. Equations 2018, No. 169, 17 pp.
    MathSciNet    
Glasnik Matematicki Home Page