On the Muskat flow

Abstract

Of concern is the motion of two fluids separated by a free interface in a porous medium, where the velocities are given by Darcy's law. We consider the case with and without phase transition. It is shown that the resulting models can be understood as purely geometric evolution laws, where the motion of the separating interface depends in a non-local way on the mean curvature. It turns out that the models are volume preserving and surface area reducing, the latter property giving rise to a Lyapunov function. We show well-posedness of the models, characterize all equilibria, and study the dynamic stability of the equilibria. Lastly, we show that solutions which do not develop singularities exist globally and converge exponentially fast to an equilibrium.

arXiv comment15 pages

References in corpus (6)

• On the global existence for the Muskat problem
• Breakdown of smoothness for the Muskat problem
• A note on stability shifting for the Muskat problem II: Stable to Unstable and back to Stable
• A note on stability shifting for the Muskat problem
• Turning waves and breakdown for incompressible flows
• The Verigin problem with and without phase transition

Cited by in corpus (5)

• The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results
• On the thin film Muskat and the thin film Stokes equations
• Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
• Weak solutions to the Muskat problem with surface tension via optimal transport
• Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data