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.
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] Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, , (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] 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] Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monogr. Math., Springer, New York (2007)
, ,[6] A justification for the thin film approximation of Stokes flow with surface tension, J. Differential Equations 245 (2008), 2802-2845 | Zbl
, ,[7] 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] Compact sets in the space
[10] The Porous Media Equation, Clarendon Press, Oxford (2007)
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