Local Classical Solution of Muskat Free Boundary Problem

Abstract

In this paper we consider the two-dimensional Muskat free boundary problem: Δu_i(x,t) = 0 in space-time domain Q_i (i = 1,2), here tis a parameter. The unknown surface Γ_pT (free boundary) is tltc common part of the boundaries of Q_1 and Q_2. The free boundary conditions are u_1(x,t) = u_2(x,t) and -k_1\frac{∂u_1}{∂n} = -k_2\frac{∂u_2}{∂n} = V_n. If the initial normal velocity of the free boundary is positive, we shall prove the existence of classical solution locally in time and uniqueness by making use of Newton's iteration method.