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 L p (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)

Cité par Sources :