Nonlinear stability of the Muskat problem with capillary pressure at the free boundary
The paper deals with the 2D Muskat problem which models the motion of two immiscible fluids (water and oil) in a porous medium. Starting from Darcy's law and from the equation of continuity, the authors prove that the potential of velocity of the two immiscible fluids satisfy Laplace's equation with appropriate conditions on the boundaries and on the moving free interface (the capillary pressure is proportional to the mean curvature of the interface and the velocity of each fluid on the interface coincides with the velocity of the interface). Considering nearly radially symmetric initial data, depending on a small parameter, the authors prove that the problem has a unique global solution which is analytic with respect to the small parameter. The solution converges to the stationary solution (which is not generally asymptotically stable) provided the initial data satisfy a sequence of necessary nonlinear constraints.
scientific article; zbMATH DE number 1901880