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This article is devoted to the study of a nonlinear and nonlocal parabolicequation introduced by Stefan Steinerberger to study the roots of polynomialsunder differentiation; it also appeared in a work by Dimitri Shlyakhtenko andTerence Tao on free convolution. Rafael Granero-Belinch\'on obtained a globalwell-posedness result for positive initial data small enough in a Wiener space,and recently Alexander Kiselev and Changhui Tan proved a global well-posednessresult for any positive initial data in the Sobolev space $H^s(\mathbb{S})$with $s>3/2$. In this paper, we consider the Cauchy problem in the criticalspace $H^{1/2}(\mathbb{S})$. Two interesting new features, at this level ofregularity, are that the equation can be written in the form $$ \partial_tu+V\partial_x u+\gamma \Lambda u=0, $$ where $\gamma$ is non-negative but notbounded from below and $V/\sqrt{\gamma}$ is not bounded. Therefore, theequation is only weakly parabolic. We prove that nevertheless the Cauchyproblem is well posed locally in time and that the solutions are smooth forpositive times. Combining this with the results of Kiselev and Tan, this givesa global well-posedness result for any positive initial data in$H^{1/2}(\mathbb{S})$. Our proof relies on sharp commutator estimates andintroduces a strategy to prove a local well-posedness result in a situationwhere the lifespan depends on the profile of the initial data and not only onits norm.

On the dynamics of the roots of polynomials under differentiation