Title: Local regularization of the one-phase Hele-Shaw flow

Authors: Sunhi Choi, David Jerison and Inwon Kim

Issue: Volume 58 (2009), Issue 6, 2765-2804

Abstract: This article presents a local regularity theorem for the one-phase Hele-Shaw flow. We prove that if the Lipschitz constant of the initial free boundary in a unit ball is small, then for small uniform positive time the solution is smooth. This result improves on our earlier results in [S. Choi, D.S. Jerison, and I.C. Kim, \emph{Regularity for the one-phase Hele-Shaw problem from a Lipschitz initial surface}, Amer. J. Math. \textbf{129} (2007), 527--582] because it is scale-invariant. As a consequence we obtain existence, uniqueness and regularity properties of global solutions with Lipschitz initial free boundary.