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 u_t=Δu^m, m>1, with initial data u₀ nonnegative, integrable, and compactly supported. We show that if the initial pressure f₀=u₀^m− 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 f^m−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.