A Cauchy integral approach to Hele-Shaw problems with a free boundary: The case of zero surface tension

In this paper, we study a nonlinear and nonlocal free-boundary dynamics - the Hele-Shaw problem without surface tension when the fluid domain is either bounded or unbounded. The key idea is to use a global quantity, the Cauchy integral of the free boundary, to capture the motion of the boundary. This Cauchy integral is shown to be linear in time. The free boundary at a fixed time is then recovered from its Cauchy integral at that time. The main tool in our analysis is CHEREDNICHENKO'S theorem concerning the inverse properties of the Cauchy integrals. As products of our approach, we establish the short-time existence and uniqueness of classical solutions for analytic initial boundaries. We also show the nonexistence of classical solutions for all smooth but non-analytic initial boundaries when there is a sink at either a finite point or at infinity. When the fluid domain is bounded, all solutions except the circular one break down before all the fluid is sucked out from the sink. Regularity results are also obtained when there is a source at a finite point or at infinity.