Well-posedness of the Muskat problem with $H^2$ initial data

Abstract

We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small $H^2$ perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for $H^2$ initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.

49 pages

References in corpus (2)

• Absence of splash singularities for SQG sharp fronts and the Muskat problem
• Splash singularities for the one-phase Muskat problem in stable regimes

Cited by in corpus (1)

• Non-splat singularity for the one-phase Muskat problem