All time C∞-regularity of the interface in degenerate diffusion: a geometric approach

Abstract

We study the connection between the geometry and all time regularity of the interface in degenerated diffusion. Our model considers the porous medium equation ut=Δum, m>1, with initial data u0 nonnegative, integrable, and compactly supported. We show that if the initial pressure f0=u0m− is smooth up to the interface and in addition it is root-concave and also satisfies the nondegeneracy condition |Df0|≠0 at $\partial\overline {\rm supp}$f0, then the pressure fm−1 remains C∞-smooth up to the interface and root-concave, for all time $0 < t < ∞$. In particular, the free boundary is C∞-smooth for all time.