Nonlinear stability of the Muskat problem with capillary pressure at the free boundary Original Research Article

The Muskat problem models the motion of two immiscible fluids in a porous medium. We assume that the medium occupies the exterior of a circle r=δ, that the fluids are incompressible, and that the capillary pressure at the interface is nonnegligible. We take any radially symmetric stationary solution with interface r=Rs,Rs>δ, and consider the Muskat problem for an initial interface r=Rs+ελ0(θ), |ε| small. We prove that this problem has a unique global solution which is analytic in ε and which converges to the stationary solution, provided λ0 satisfies a sequence of nonlinear constraints. These constraints are satisfied in the case where View the MathML source any integer ⩾2. The nonlinear constraints are, under some assumptions on the solution, also necessary.