Stabilization of free surface flows in periodic porous media

The flow of an incompressible Newtonian fluid in a periodic porous medium is studied. The equations governing this motion form a free boundary problem involving a potential flow problem and the evolution of a free surface. The flow is driven by gravitational forces and it is stabilized by an external source. The existence of a unique classical solution for large initial data is proved in the sense that there is an unbounded set of initial conditions for which a unique classical solution exists.