Gravity-driven nonlinear flow in areas with impermeable inclusions

Free-surface flows of a fluid in a porous layer containing impermeable inclusions on the boundary of the flow region are studied. The solvability of these problems in the case of a nonlinear constitutive equation are proved such as various special cases of free-surface fluid dynamics problems in multiply connected domains, flow in an unbounded porous layer with unknown interfaces and a compactly-supported inclusion, boundary value problem, A priori estimates, construction of the operator of the problem, solvability of the problem, and multiply connected domains.