Opuscula Math. 42, no. 1 (2022), 5-29
https://doi.org/10.7494/OpMath.2022.42.1.5

 
Opuscula Mathematica

Uniqueness of solution of a nonlinear evolution dam problem in a heterogeneous porous medium

Messaouda Ben Attia
Elmehdi Zaouche
Mahmoud Bousselsal

Abstract. By choosing convenient test functions and using the method of doubling variables, we prove the uniqueness of the solution to a nonlinear evolution dam problem in an arbitrary heterogeneous porous medium of \(\mathbb{R}^n\) (\(n\in \{2,3\}\)) with an impermeable horizontal bottom.

Keywords: test function, method of doubling variables, nonlinear evolution dam problem, heterogeneous porous medium, uniqueness.

Mathematics Subject Classification: 35A02, 76S05.

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  1. R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  2. S.J. Alvarez, Problemas de frontera libre en teoria de lubrificacion, Ph.D. Thesis, Universidad Complutense de Madrid, Spain, 1986.
  3. S.J. Alvarez, J. Carrillo, A free boundary problem in theory of lubrication}, Comm. Partial Differential Equations 19 (1994), 1743-1761.
  4. S.J. Alvarez, R. Oujja, On the uniqueness of the solution of an evolution free boundary problem in theory of lubrication, Nonlinear Anal. 54 (2003), 845-872.
  5. F. Andreu, J.M. Mazón, J.S. Moll, The total variation flow with nonlinear boundary conditions, Asymptot. Anal. 43 (2005), 9-46.
  6. Ph. Bénilan, J. Carrillo, P. Wittbold, Renormalized entropy solutions of scalar conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (2000), 313-327.
  7. M. Bousselsal, E. Zaouche, The evolution dam problem for a compressible fluid with nonlinear Darcy's law and Dirichlet boundary condition, Math. Methods Appl. Sci. 44 (2021), 66-90.
  8. J. Carrillo, On the uniqueness of the solution of the evolution dam problem, Nonlinear Anal. 22 (1994), 573-607.
  9. J. Carrillo, A. Lyaghfouri, The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998), 453-505.
  10. J. Carrillo, P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Differential Equations 156 (1999), 93-121.
  11. E. DiBenedetto, A. Friedam, Periodic behaviour for the evolutionary dam problem and related free boundary problems Evolutionary dam problem, Comm. Partial Differential Equations 11 (1986), no. 12, 1297-1377.
  12. V.G. Jakubowski, P. Wittbold, On a nonlinear elliptic-parabolic integro-differential equation with \(L^1\)-data, J. Differential Equations 197 (2004), 427-445.
  13. K.H. Karlsen, M. Ohlberger, A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations, J. Math. Anal. Appl. 275 (2002), 439-458.
  14. S.N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 217-243.
  15. M. Lazar, D. Mitrović, Existence of solutions for a scalar conservation law with a flux of low regularity, Electron. J. Differential Equations 2016 (2016), 1-18.
  16. A. Lyaghfouri, The evolution dam problem for nonlinear Darcy's law and Dirichlet boundary conditions, Port. Math. 56 (1999), 1-38.
  17. A. Lyaghfouri, A regularity result for a heterogeneous evolution dam problem, Z. Anal. Anwend. 24 (2005), 149-166.
  18. A. Lyaghfouri, E. Zaouche, Uniqueness of solution of the unsteady filtration problem in heterogeneous porous media, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112 (2018), 89-102.
  19. F.Kh. Mukminov, Uniqueness of the renormalized solutions to the Cauchy problem for an anisotropic parabolic equation, Ufa Math. J. 8 (2016), 44-57.
  20. F.Kh. Mukminov, Uniqueness of the renormalized solution of an elliptic-parabolic problem in anisotropic Sobolev-Orlicz spaces, Sb. Math. 208 (2017), 1187-1206.
  21. S. Ouaro, H. Toure, Uniqueness of entropy solutions to nonlinear elliptic-parabolic problems, Electron. J. Differential Equations 2007 (2007), 1-15.
  22. E.Yu. Panov, On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux, J. Hyperbolic Differ. Equ. 6 (2009), 525-548.
  23. E. Zaouche, Uniqueness of solution in a rectangular domain of an evolution dam problem with heterogeneous coefficients, Electron. J. Differential Equations 2018 (2018), 1-17.
  24. E. Zaouche, Uniqueness of solution of a heterogeneous evolution dam problem associated with a compressible fluid flow through a rectangular porous medium, Glas. Mat. Ser. III 55 (2020), 93-99.
  • Messaouda Ben Attia
  • University of Kasdi Merbah, Ouargla, Laboratory of Applied Mathematics, B.P. 511, Ouargla 30000, Algeria
  • Elmehdi Zaouche (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-7777-9284
  • University of El Oued, Department of Mathematics, Labo. Oper. Theo. and Part. Diff. Eq.: Foundations and Applications, B.P. 789, El Oued 39000, Algeria
  • Mahmoud Bousselsal
  • Ecole Normale Supérieure, Department of Mathematics, Laboratoire des équations aux dérivées partielles et H.M., 16050, Vieux-Kouba, Algiers, Algeria
  • Communicated by J.I. Díaz.
  • Received: 2021-01-27.
  • Revised: 2021-10-15.
  • Accepted: 2021-10-18.
  • Published online: 2022-01-20.
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Cite this article as:
Messaouda Ben Attia, Elmehdi Zaouche, Mahmoud Bousselsal, Uniqueness of solution of a nonlinear evolution dam problem in a heterogeneous porous medium, Opuscula Math. 42, no. 1 (2022), 5-29, https://doi.org/10.7494/OpMath.2022.42.1.5

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