Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 583-598.

We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.

DOI : 10.1016/j.anihpc.2011.04.001
Classification : 35K65, 35K40, 35D30, 35B35, 35Q35
Mots-clés : Degenerate parabolic system, Weak solutions, Exponential stability, Thin film, Liapunov functional
@article{AIHPC_2011__28_4_583_0,
     author = {Escher, Joachim and Lauren\c{c}ot, Philippe and Matioc, Bogdan-Vasile},
     title = {Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {583--598},
     publisher = {Elsevier},
     volume = {28},
     number = {4},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.04.001},
     zbl = {1227.35177},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.04.001/}
}
TY  - JOUR
AU  - Escher, Joachim
AU  - Laurençot, Philippe
AU  - Matioc, Bogdan-Vasile
TI  - Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
SP  - 583
EP  - 598
VL  - 28
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2011.04.001/
DO  - 10.1016/j.anihpc.2011.04.001
LA  - en
ID  - AIHPC_2011__28_4_583_0
ER  - 
%0 Journal Article
%A Escher, Joachim
%A Laurençot, Philippe
%A Matioc, Bogdan-Vasile
%T Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 583-598
%V 28
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2011.04.001/
%R 10.1016/j.anihpc.2011.04.001
%G en
%F AIHPC_2011__28_4_583_0
Escher, Joachim; Laurençot, Philippe; Matioc, Bogdan-Vasile. Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 583-598. doi : 10.1016/j.anihpc.2011.04.001. http://www.numdam.org/articles/10.1016/j.anihpc.2011.04.001/

[1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, H. Schmeisser, H. Triebel (ed.), Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math. vol. 133, Teubner, Stuttgart/Leipzig (1993), 9-126

[2] J. Escher, M. Hillairet, Ph. Laurençot, C. Walker, Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant, Indiana Univ. Math. J., in press.

[3] J. Escher, A.-V. Matioc, B.-V. Matioc, A generalised Rayleigh–Taylor condition for the Muskat problem, arXiv:1005.2511v1 | Zbl

[4] J. Escher, A.-V. Matioc, B.-V. Matioc, Modelling and analysis of the Muskat problem for thin fluid layers, J. Math. Fluid Mech., doi:10.1007/s00021-011-0053-2, in press.

[5] I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monogr. Math., Springer, New York (2007)

[6] M. Günther, G. Prokert, A justification for the thin film approximation of Stokes flow with surface tension, J. Differential Equations 245 (2008), 2802-2845 | Zbl

[7] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel (1995) | Zbl

[8] B.-V. Matioc, G. Prokert, Hele-Shaw flow in thin threads: A rigorous limit result, preprint.

[9] J. Simon, Compact sets in the space Lp(0,T;B), Ann. Mat. Pura Appl. 4 no. 146 (1987), 65-96 | Zbl

[10] J.L. Vázquez, The Porous Media Equation, Clarendon Press, Oxford (2007)

  • Laurençot, Philippe; Matioc, Bogdan-Vasile Bounded weak solutions to a class of degenerate cross-diffusion systems, Annales Henri Lebesgue, Volume 6 (2023), pp. 847-874 | DOI:10.5802/ahl.179 | Zbl:1523.35212
  • Laurençot, Philippe; Matioc, Bogdan-Vasile The porous medium equation as a singular limit of the thin film Muskat problem, Asymptotic Analysis, Volume 131 (2023) no. 2, pp. 255-271 | DOI:10.3233/asy-221774 | Zbl:1509.35228
  • Wu, Tao Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem, Open Mathematics, Volume 21 (2023), p. 12 (Id/No 20220590) | DOI:10.1515/math-2022-0590 | Zbl:1526.35226
  • Li, Jia; Bi, Changchun Study of weak solutions of variational inequality systems with degenerate parabolic operators and quasilinear terms arising Americian option pricing problems, AIMS Mathematics, Volume 7 (2022) no. 11, p. 19758 | DOI:10.3934/math.20221083
  • Laurençot, Philippe; Matioc, Bogdan-Vasile Bounded weak solutions to the thin film Muskat problem via an infinite family of Liapunov functionals, Transactions of the American Mathematical Society, Volume 375 (2022) no. 8, pp. 5963-5986 | DOI:10.1090/tran/8688 | Zbl:1496.35239
  • Gancedo, Francisco; Granero-Belinchón, Rafael; Scrobogna, Stefano Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 37 (2020) no. 6, pp. 1299-1343 | DOI:10.1016/j.anihpc.2020.04.005 | Zbl:1459.76053
  • Bruell, Gabriele; Granero-Belinchón, Rafael On a thin film model with insoluble surfactant, Journal of Differential Equations, Volume 268 (2020) no. 12, pp. 7582-7608 | DOI:10.1016/j.jde.2019.11.080 | Zbl:1473.35102
  • Oulhaj, Ahmed Ait Hammou; Maltese, David Convergence of a positive nonlinear control volume finite element scheme for an anisotropic seawater intrusion model with sharp interfaces, Numerical Methods for Partial Differential Equations, Volume 36 (2020) no. 1, pp. 133-153 | DOI:10.1002/num.22422 | Zbl:1452.65246
  • Oulhaj, Ahmed Ait Hammou; Cancès, Clément; Chainais-Hillairet, Claire; Laurençot, Philippe Large time behavior of a two phase extension of the porous medium equation, Interfaces and Free Boundaries, Volume 21 (2019) no. 2, pp. 199-229 | DOI:10.4171/ifb/421 | Zbl:1423.35225
  • Bruell, Gabriele; Granero-Belinchón, Rafael On the thin film Muskat and the thin film Stokes equations, Journal of Mathematical Fluid Mechanics, Volume 21 (2019) no. 2, p. 31 (Id/No 33) | DOI:10.1007/s00021-019-0437-2 | Zbl:1417.35057
  • Kim, Inwon; Mészáros, Alpár Richárd On nonlinear cross-diffusion systems: an optimal transport approach, Calculus of Variations and Partial Differential Equations, Volume 57 (2018) no. 3, pp. 1-40 (Id/No 79) | DOI:10.1007/s00526-018-1351-9 | Zbl:1393.35118
  • Alkhayal, Jana; Issa, Samar; Jazar, Mustapha; Monneau, Régis Existence result for degenerate cross-diffusion system with application to seawater intrusion, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1735-1758 | DOI:10.1051/cocv/2017058 | Zbl:1410.35106
  • Oulhaj, Ahmed Ait Hammou Numerical analysis of a finite volume scheme for a seawater intrusion model with cross-diffusion in an unconfined aquifer, Numerical Methods for Partial Differential Equations, Volume 34 (2018) no. 3, pp. 857-880 | DOI:10.1002/num.22234 | Zbl:1407.76087
  • Oulhaj, Ahmed Ait Hammou A Finite Volume Scheme for a Seawater Intrusion Model with Cross-Diffusion, Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects, Volume 199 (2017), p. 421 | DOI:10.1007/978-3-319-57397-7_35
  • Laurençot, Philippe; Matioc, Bogdan-Vasile Self-similarity in a thin film Muskat problem, SIAM Journal on Mathematical Analysis, Volume 49 (2017) no. 4, pp. 2790-2842 | DOI:10.1137/16m1055335 | Zbl:1386.35238
  • Matioc, Bogdan-Vasile Non-uniform continuity of the semiflow map associated to the porous medium equation, Bulletin of the London Mathematical Society, Volume 46 (2014) no. 6, pp. 1145-1155 | DOI:10.1112/blms/bdu068 | Zbl:1307.35032
  • Escher, Joachim; Matioc, Bogdan-Vasile Non-negative global weak solutions for a degenerated parabolic system approximating the two-phase Stokes problem, Journal of Differential Equations, Volume 256 (2014) no. 8, pp. 2659-2676 | DOI:10.1016/j.jde.2014.01.005 | Zbl:1288.35291
  • Laurençot, Philippe; Matioc, Bogdan-Vasile A thin film approximation of the Muskat problem with gravity and capillary forces, Journal of the Mathematical Society of Japan, Volume 66 (2014) no. 4, pp. 1043-1071 | DOI:10.2969/jmsj/06641043 | Zbl:1307.35137
  • Laurençot, Philippe; Matioc, Bogdan-Vasile A gradient flow approach to a thin film approximation of the Muskat problem, Calculus of Variations and Partial Differential Equations, Volume 47 (2013) no. 1-2, pp. 319-341 | DOI:10.1007/s00526-012-0520-5 | Zbl:1264.35129
  • Escher, Joachim; Matioc, Bogdan-Vasile Existence and stability of solutions for a strongly coupled system modelling thin fluid films, NoDEA. Nonlinear Differential Equations and Applications, Volume 20 (2013) no. 3, pp. 539-555 | DOI:10.1007/s00030-012-0166-1 | Zbl:1268.76005
  • Escher, Joachim; Matioc, Anca-Voichita; Matioc, Bogdan-Vasile Thin-film approximations of the two-phase Stokes problem, Nonlinear Analysis: Theory, Methods Applications, Volume 76 (2013), p. 1 | DOI:10.1016/j.na.2012.07.034
  • Escher, Joachim; Matioc, Anca-Voichita; Matioc, Bogdan-Vasile Modelling and analysis of the Muskat problem for thin fluid layers, Journal of Mathematical Fluid Mechanics, Volume 14 (2012) no. 2, pp. 267-277 | DOI:10.1007/s00021-011-0053-2 | Zbl:1294.76235

Cité par 22 documents. Sources : Crossref, zbMATH