Variatonal formulation for the lubrication approximation of the Hele-Shaw flow

It has been recently discovered that both the surface tension driven one-ph ase Hele-Shaw flow and its lubrication approximation can be understood as ( continuous limits of time-discretized) gradient flows of the corresponding surface energy functionals with respect to the Wasserstein metric. Here we complete the connection between the two problems, proving that the time-dis cretized lubrication approximation is the F-limit of suitably rescaled time -discretized Hele-Shaw flows in half space.