Variational formulation for the lubrication approximation of the Hele-Shaw flow
Lorenzo Giacomelli and Felix Otto
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Submission date: 09. May. 2000
published in: Calculus of variations and partial differential equations, 13 (2001) 3, p. 377-403
DOI number (of the published article): 10.1007/s005260000077
MSC-Numbers: 35A15, 49J45, 35K25, 35K55, 35K65, 35R35, 76D27, 76D08, 76A20
Keywords and phrases: variational methods, gamma-convergence, higher order nonlinear degenerate parabolic equations, free boundary problems, hele-shaw flow, lubrication theory, thin fluid films
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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 -limit of suitably rescaled time-discretized Hele-Shaw flows in half space.