Long time regularity of solutions of the Hele-Shaw problem

In this paper we study the long-time behavior of solutions of the one phase Hele-Shaw problem without surface tension. We show that after a finite time solutions of the Hele-Shaw problem become Lipschitz continuous with nondegenerate free boundary speed. It then follows that after a finite time the solution and the free boundary become smooth in space and time. 0 Introduction Let K = {x ∈ IR : |x| = 1} and suppose that a bounded domain Ω contains K and let Ω0 = Ω − K and Γ0 = ∂Ω (see figure 1). Note that ∂Ω0 = Γ0∪∂K. Let u0 be the harmonic function in Ω0 with u0 = f ≡ 1 > 0 on K and zero on Γ0. In addition we suppose u0 satisfies (I) |Du0| > 0 on Γ0.