Concavity of Solutions of the Porous Medium Equation.

The flow of a gas through a porous medium is governed by a degenerate quasilinear parabolic equation. It is known that the nonnegative solutions to this equation possess a lower bound for the second derivative of the pressure in the spatial variables. This bound plays an important role in the mathematical treatment and is related to the entropy of the flow. Since the solutions exhibit interfaces across which v sub x jumps positively, no upper bound is possible globally for v subxx. Nevertheless it is proven that the concavity of v.,t in the region where v is positive is preserved in time. This is in itself an interesting geometric property of the solution. It also allows one to obtain precise information about the asymptotic behaviour of the flow.