Regularity of the free boundary for the porous medium equation

Abstract

We study the regularity of the free boundary for solutions of the porous medium equation u t = Δ u m u_{t}=\Delta u^{m} , m > 1 m >1 , on R 2 × [ 0 , T ] {\mathcal {R}}^{2} \times [0,T] , with initial data u 0 = u ( x , 0 ) u^{0}=u(x,0) nonnegative and compactly supported. We show that, under certain assumptions on the initial data u 0 u^{0} , the pressure f = m u m − 1 f=m\, u^{m-1} will be smooth up to the interface Γ = ∂ { u > 0 } \Gamma = \partial \{ u >0 \} , when 0 > t ≤ T 0>t\leq T , for some T > 0 T >0 . As a consequence, the free-boundary Γ \Gamma is smooth.