In this work we study the nonlocal transport equation derived recently by Steinerberger when studying how the distribution of roots of a polynomial behaves under iterated differentation of the function. 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, Cordoba, Cordoba and Fontelos.

中文翻译：

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关于描述微分下多项式根的非局部微分方程

在这项工作中，我们研究了 Steinerberger 最近在研究多项式的根分布在函数的迭代微分下的表现时导出的非局部传输方程。特别是，我们研究了方程的适定性，建立了解的一些定性性质，并给出了确保弱解和强解全局存在的条件。最后，我们提出了 Steinerberger 获得的方程与 Chae、Cordoba、Cordoba 和 Fontelos 使用的表面准地转方程的一维模型之间的联系。