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

TL;DR

This paper analyzes the long-term behavior of a two-phase porous medium equation modeling seawater intrusion, identifying unique steady states and proving convergence of solutions to these states over time.

Contribution

It characterizes the unique radially symmetric steady states as energy minimizers and proves the convergence of solutions to these states as time approaches infinity.

Findings

Unique steady states are radially symmetric and energy minimizers.

Solutions converge to the stationary state as time tends to infinity.

Numerical simulations illustrate stationary states and convergence rates.

Abstract

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.