All Time Smooth Solutions of the One-Phase Stefan Problem and the Hele-Shaw Flow

Abstract

Abstract We consider the one phase free boundary problem of Stefan (or Hele-Shaw) type: find {u, Ω} such that Ω = {u > 0} and with Q T = ℝ n × (0, T), T > 0. Under the condition that u o is C 1, 1 and log u o is concave on Ω o , we show that the concavity of log u(⋅, t) is preserved under the flow. As a consequence, we show that there exists a solution which is smooth at all time, up to the interface. In particular, the interface is smooth.