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 dynamic s - the Hele-Shaw problem without surface tension when the fluid domai n is either bounded or unbounded. The key idea is to use a global quan tity, the Cauchy integral of the free boundary, to capture the motion of the boundary. This Cauchy integral is shown to be linear in time. T he free boundary at a fixed time is then recovered from its Cauchy int egral 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 un iqueness of classical solutions for analytic initial boundaries. We al so show the nonexistence of classical solutions for all smooth but non -analytic initial boundaries when there is a sink at either a finite p oint or at infinity. When the fluid domain is bounded, all solutions e xcept the circular one break down before all the fluid is sucked out f rom the sink. Regularity results are also obtained when there is a sou rce at a finite point or at infinity.