*Submitted Paper*

**Inserted:** 21 apr 2021

**Year:** 2021

**Abstract:**

This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by
Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work
by Dimitri Shlyakhtenko and Terence Tao on free convolution.
Rafael Granero-Belinch\'on obtained a global well-posedness result for initial data small enough in a Wiener space, and recently
Alexander Kiselev and Changhui Tan proved a global well-posedness result for any initial data in the Sobolev space $H^s(S)$ with $s>3/2$.
In this paper, we consider the Cauchy problem in the critical space $H^{1/2}(S)$. Two interesting new features, at this level of regularity, are that
the equation can be written in the form
$$
\partial_{t} u+V\partial_{x} u+\gamma \Lambda u=0,
$$
where $V$ is not bounded and $\gamma$ is not bounded from below. Therefore, the equation is only weakly parabolic.
We prove that nevertheless the Cauchy problem is well posed locally in time and that the solutions are smooth for positive times.
Combining this with the results of Kiselev and Tan, this gives a global well-posedness result for any initial data in $H^{1/2}(S)$.
Our proof relies on sharp commutators estimates and introduces a strategy to prove a local well-posedness result in a situation
where the lifespan depends on the profile of the initial data and not only on its norm.