We rigorously prove the convergence of appropriately scaled solutions of the 2D Hele–Shaw moving boundary problem with surface tension in the limit of thin threads to the solution of the formally corresponding Thin Film equation. The proof is based on scaled parabolic estimates for the nonlocal, nonlinear evolution equations that arise from these problems.
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Bogdan-Vasile Matioc, Georg Prokert, Hele–Shaw flow in thin threads: A rigorous limit result. Interfaces Free Bound. 14 (2012), no. 2, pp. 205–230