Global classical solution of Muskat free boundary problem ✩

In this paper the Muskat problem which describes a two-phase flow of two fluids, for example, oil and water, in porous media is discussed. The problem involves in seeking two time-dependent harmonic functions u1(x,y, t) and u2(x,y, t) in oil and water regions, respectively, and the interface between oil and water, i.e., the free boundary Γ : y = ρ(x, t), such that on the free boundary u1 = u2, Vn =−k1 ∂u1 ∂n =−k2 ∂u2 ∂n , where n the unit normal vector on the free boundary toward oil region, Vn is the normal velocity of the free boundary Γ , k1 and k2 are positive constants satisfying k1 > k2. We prove the existence of classical solution globally in time under some reasonable assumptions. The argument developed in this paper can be used in any multidimensional case.  2003 Elsevier Inc. All rights reserved