Non-preservation of α-concavity for the porous medium equation · arXivDesk
2011.03063Nov 5, 202015 pages, 1 figure; v2 this is a substantially revised version containing a new result on interior concavity breaking which in particular resolves an open problem of Vázquez; v3 final version, typos corrected, to appear in Adv. Math
Non-preservation of α-concavity for the porous medium equation
We show that the porous medium equation does not in general preserve α-concavity of the pressure for 0≤α<1/2 or 1/2<α≤1. In particular, this resolves an open problem of Vázquez 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
Nearby in the stack
α>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≤α<1/2
, we give another set of examples to show that concavity can be broken at a boundary point.
arXiv did not return neighbours for this paper. Try again in a bit.