Non-preservation of $\alpha$-concavity for the porous medium equation

TL;DR

This paper demonstrates that the porous medium equation generally does not preserve $eta$-concavity of pressure for certain ranges of $eta$, resolving an open problem and providing explicit examples of instantaneously broken concavity.

Contribution

It proves that $eta$-concavity is not preserved for $0 extlesseta extless 1/2$ and $1/2 extlesseta extless 1$, resolving a longstanding open problem and sharpening previous results.

Findings

Concavity can be instantaneously broken at interior points.

Concavity can be broken at boundary points for certain $eta$.

The results are sharp, confirming preservation only at $eta=1/2$.

Abstract

We show that the porous medium equation does not in general preserve $\alpha$-concavity of the pressure for $0\le\alpha<1/2$ or $1/2<\alpha\le 1$. In particular, this resolves an open problem of V\'azquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small $\alpha>0$. Since Daskalopoulos-Hamilton-Lee showed that $1/2$-concavity is preserved, our result is sharp. Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For $0\le\alpha<1/2$, we give another set of examples to show that concavity can be broken at a boundary point.