All time C-infinity-regularity of the interface in degenerate diffusion: A geometric approach

摘要

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) = Deltau(m), m > 1, with initial data u(0) nonnegative, integrable, and compactly supported. We show that if the initial pressure f(0) = u(0)(m-1) is smooth up to the interface and in addition it is root-concave and also satisfies the nondegeneracy condition /Df(0)/ not equal 0 at partial derivative(supp) over barf(0), then the pressure f = u(m-1) remains C-infinity-smooth up to the interface and root-concave, for all time 0 < t < infinity. In particular the free boundary is C-infinity-smooth for all time. [References: 10]