Local regularization of the one-phase hele-shaw flow

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 [4] because it is scaleinvariant. As a consequence we obtain existence, uniqueness and regularity properties of global solutions with Lipschitz initial free boundary.