2020/11/05 by Albert Chau, Chau, Albert, Ben Weinkove +1 · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2011.03063
openalex publication_date 2020/11/05 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/28
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 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.