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MiS Preprint

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

Lorenzo Giacomelli and Felix Otto


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.

MSC Codes:
35A15, 49J45, 35K25, 35K55, 35K65, 35R35, 76D27, 76D08, 76A20
variational methods, gamma-convergence, higher order nonlinear degenerate parabolic equations, free boundary problems, hele-shaw flow, lubrication theory, thin fluid films

Related publications

2001 Repository Open Access
Lorenzo Giacomelli and Felix Otto

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

In: Calculus of variations and partial differential equations, 13 (2001) 3, pp. 377-403