Local regularization of the one-phase Hele-Shaw flow Sunhi ChoiDavid JerisonInwon Kim 35M10Hele-Shaw flowfree boundaryregularityviscosity solutions 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. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3802 10.1512/iumj.2009.58.3802 en Indiana Univ. Math. J. 58 (2009) 2765 - 2804 state-of-the-art mathematics http://iumj.org/access/