Flatness implies smoothness for solutions of the porous medium equation

Abstract

One of the major problems in the theory of the porous medium equation $\partial _t\rho =\Delta _x\rho ^m,\,m > 1$, is the regularity of the solutions $\rho (t,x)\ge 0$ and the free boundaries $\Gamma =\partial \{(t,x): \rho >0\}$. Here we assume flatness of the solution and derive $C^\infty$ regularity of the interface after a small time, as well as $C^\infty$ regularity of the solution in the positivity set and up to the free boundary for some time interval. The proof starts from Caffarelli’s blueprint of an improvement of flatness by rescaling, and combines it with the Carleson measure approach applied to the degenerate subelliptic equation satisfied by the pressure of the porous medium equation in transformed coordinates. The improvement of flatness finally hinges on Gaussian estimates for the subelliptic problem. We use these facts to prove the following eventual regularity result: solutions defined in the whole space with compactly supported initial data are smooth after a finite time $T_r$ that depends on $\rho _0$. More precisely, we prove that for $t \ge T_r$ the pressure $\rho ^{m-1}$ is $C^\infty$ in the positivity set and up to the free boundary, which is a $C^\infty$ hypersurface. Moreover, $T_r$ can be estimated in terms of only the initial mass and the initial support radius. This regularity result eliminates the assumption of non-degeneracy on the initial data that has been carried on for decades in the literature. Let us recall that regularization for small times is false, and that as $t\rightarrow \infty$ the solution increasingly resembles a Barenblatt function and the support looks like a ball.