A filtration problem with nonlinear Darcy’s law and generalized boundary conditions.

Abstract

This article studies a filtration problems with nonlinear Darcy’s law, e.g. ~v = −|rp|q−2rp,
where p is the fluid pressure , q > 1, and ~v is the velocity, governed by the mass conservation
law div(~v) = 0. This leads to a free boundary problem since the movement of the fluid takes place only in the “wet part” of the domain which is also unknown. The authors give a unified formulation of the problem to include general boundary conditions,
such as Dirichlet, Neumann, and so-called leaky boundary conditions. The existence of (weak) solutions is established, for both bounded and unbounded domains, by an
approximation method.