Communications in Mathematical Sciences

Volume 18 (2020)

Number 6

On a nonlocal differential equation describing roots of polynomials under differentiation

Pages: 1643 – 1660



Rafael Granero-Belinchón (Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain)


In this work we study the nonlocal transport equation derived recently by Steinerberger [Proc. Amer. Math. Soc., 147(11):4733–4744, 2019]. When this equation is considered on the real line, it describes how the distribution of roots of a polynomial behaves under iterated differentiation of the function. This equation can also be seen as a nonlocal fast diffusion equation. In particular, we study the well-posedness of the equation, establish some qualitative properties of the solution and give conditions ensuring the global existence of both weak and strong solutions. Finally, we present a link between the equation obtained by Steinerberger and a one-dimensional model of the surface quasi-geostrophic equation used by Chae et al. [Adv. Math., 194(1):203–223, 2005].


nonlocal fast diffusion, global existence, maximum principle

2010 Mathematics Subject Classification

35B65, 35K55, 35K65, 35S30

Received 4 November 2019

Accepted 30 March 2020

Published 4 November 2020