Two-phase flows involving capillary barriers in heterogeneous porous media

We consider a simplified model of a two-phase flow through a heterogeneous porous medium, in which convection is neglected. This leads to a nonlinear degenerate parabolic problem in a domain divided into an arbitrary finite number of homogeneous porous media. We introduce a new way to connect capillary pressures on the interfaces between the homogeneous domains, which leads to a general notion of solution. We then compare this notion of solution with an existing one, showing that it allows one to deal with a larger class of problems. We prove the existence of such a solution in a general case, and the existence and uniqueness of a regular solution in the one-dimensional case for initial data regular enough.