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We consider the fluid interface problem given by two incompressible fluidswith different densities evolving by Darcy's law. This scenario is known as theMuskat problem for fluids with the same viscosities, being in two dimensionsmathematically analogous to the two-phase Hele-Shaw cell. We prove in thestable case (the denser fluid is below) a maximum principle for the $L^\infty$norm of the free boundary.

A maximum principle for the Muskat problem for fluids with different densities