Regularity of the free boundary for the porous medium equation

We study the regularity of the free boundary for solutions of the porous medium equation $u_{t}=\Delta u^{m}$, $m >1$, on ${\mathcal{R}}^{2} \times [0,T]$, with initial data $u^{0}=u(x,0)$ nonnegative and compactly supported. We show that, under certain assumptions on the initial data $u^{0}$, the pressure $f=m\, u^{m-1}$ will be smooth up to the interface $\Gamma = \partial \{ u >0 \}$, when $0<t\leq T$, for some $T >0$. As a consequence, the free-boundary $\Gamma$ is smooth.