Variatonal formulation for the lubrication approximation of the Hele-Shaw flow
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It has been recently discovered that both the surface tension driven one-phase 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-discretized lubrication approximation is the \(\Gamma\)-limit of suitably rescaled time-discretized Hele-Shaw flows in half space.
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