Flatness implies smoothness for the porous medium equation
Seminar Room 2, Newton Institute Gatehouse
We call a solution to the porous medium equation flat on a unit size space time cylinder if it is sandwiched between two close and parallel traveling wave solutions of velocity 1.
Those solutions are smooth in the sense that the pressure is a smooth function on the closure of its support, and inside a smaller space time cylinder, with nonvanishing gradient at the boundary of the support. The proof follows Caffarellis strategy of improving the flatness for rescaled solutions. An important part consists in a linearization of equation and geometry.