带有重力项的一维渗流方程解的自由边界正则性

Regularity of the Free Boundary for the Solution of the Porous Medium Equation with the Grarity Term in One-Dimension

本文讨论带重力项的一维渗流方程解的自由边界正则性,正如我们所知此类方程当初值具有紧支集,则存在两条Lipschitz连续、单调的曲线ζ_i(t)(i=1,2)称之为解的自由边界:当且仅当ζ_1(t)<x<ζ_2(t),0<t<+∞,时解u(x,t)>0;我们在本文进一步讨论ζ_i(t)的性质,在对m、n关系作一些假设后,证明了除了可能有一点外ζ_i(t)(i=1,2)是连续可微的,同时分别给出解在自由边界铅直处和斜处的性态。

In this paper we consider the regularity of the free boundary for the one-dimensional flow in a vertical column porous medium. As is known to all, there exist two curves called free boundaries when the initial data has compact support. We want to discuss them further. Under some hypothesis between m and n (where m, n are two parameters), we prove they are continuously differentiable except one point.