Large time behavior of a two phase extension of the porous medium equation

Ahmed Ait Hammou Oulhaj (1, 2) , Clément Cancès (2, 1) , Claire Chainais-Hillairet (1, 2) , Philippe Laurençot (3)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 IMT - Institut de Mathématiques de Toulouse UMR5219

We study the large time behavior of the solutions to a two phase extension of the porous medium equation, which models the so-called seawater intrusion problem. The goal is to identify the self-similar solutions that correspond to steady states of a rescaled version of the problem. We fully characterize the unique steady states that are identified as minimizers of a convex energy and shown to be radially symmetric. Moreover, we prove the convergence of the solution to the time-dependent model towards the unique stationary state as time goes to infinity. We finally provide numerical illustrations of the stationary states and we exhibit numerical convergence rates.

Mots clés: Muskat problem; Cross-diffusion system; Two-phase porous media flows; large time behavior

Domaines: Equations aux dérivées partielles [math.AP]

Ahmed Ait Hammou Oulhaj, Clément Cancès, Claire Chainais-Hillairet, Philippe Laurençot. Large time behavior of a two phase extension of the porous medium equation. Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, 2019, 21, pp.199-229. ⟨10.4171/IFB/421⟩. ⟨hal-01752759⟩