Eventual C^∞-Regularity and Concavity for Flows in One-Dimensional Porous Media

Abstract

We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation

$${u_t} = {eft( {{u^m}} ight)_{xx}}inQ = R imes eft( {0,nfty } ight),$$

$$ueft( {x,0} ight) = {u_0}eft( x ight)forx n R,$$

with m >1 and, u_o a continuous, nonnegative function.

It is well known that, across a moving interface $x = eta eft( t ight)$ of the solution u(x,t), the derivatives υ_t and υ_χ of the pressure $v = eft( {m/eft( {m - 1} ight)} ight){u^{m - 1}}$ have jump discontinuities. We prove that each moving part of the interface is a C ^∞-curve and that υ is C ^∞ on each side of the moving interface (and up to it). We also prove that for solutions with compact support the pressure becomes a concave function of x after a finite time. This fact implies sharp convergence rates for the solution and the interfaces as t → ∞.