Regularity for the Porous Medium Equation

As is well known, the porous medium equation

$\displaystyle u_t = \left(u^m\right)_{xx},\quad x\in\mathbb{R},\,t > 0, \qquad (m>1) $

has solutions with finite speed of propagation. More precisely, if the initial data are nonnegative with support on a bounded interval, this remains true for all $ t>0$, i.e., $ \operatorname{supp}u(\cdot,t)=D(t)=[s_-(t),s_+(t)]$.



Standard parabolic regularity does not imply smoothness of $ v:=u^{m-1}$ up to the free boundaries $ s_\pm$. However, if $ v(\cdot,0)$ is continuously differentiable on $ D(0)$ with $ v_x(s_\pm(0),0)\ne0$, then $ s_\pm$ and $ v(\cdot,t)$, $ t>0$, are infinitely differentiable on $ (0,\infty)$ and $ D(t)$ respectively.


$ \square$ with H.-O. Kreiss, Mathematische Zeitschrift 192 (1986), 217-224.

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