<p>On the dynamics of the roots of polynomials under differentiation</p>

doi:10.1016/j.matpur.2022.04.001

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	On the dynamics of the roots of polynomials under differentiation
	Alazard, Thomas 1,2; Lazar, Omar 3; Nguyen, Quoc Hung 4
	2022-06-01
发表期刊	JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
ISSN	0021-7824
卷号	162 页码:1-22
摘要	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-Belinchon 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 H1/2(S). Two interesting new features, at this level of regularity, are that the equation can be written in the form & nbsp;& part;tu + V & part;(x)u +gamma lambda u = 0,& nbsp;where V is not bounded and & nbsp;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. (C)& nbsp;2022 Elsevier Masson SAS. All rights reserved.
关键词	Transport equations Nonlocal operators Well-posedness Regularity
DOI	10.1016/j.matpur.2022.04.001
收录类别	SCI
语种	英语
资助项目	French National Research Agency (ANR)[ANR-18-CE40-0027] ; Academy of Mathematics and Systems Science, Chinese Academy of Sciences startup fund ; National Natural Science Foundation[12050410257]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied ; Mathematics
WOS记录号	WOS:000798180300001
出版者	ELSEVIER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/61434
专题	中国科学院数学与系统科学研究院
通讯作者	Lazar, Omar
作者单位	1.Univ Paris Saclay, Ctr Borelli UMR9010, CNRS, ENS Paris-Saclay, Ave Sci, F-91190 Gif sur yvette, France 2.Univ Calif, Dept Math, Berkeley, CA 94720 USA 3.Amer Univ Middle East, Coll Engn & Technol, Egaila, Kuwait 4.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Alazard, Thomas,Lazar, Omar,Nguyen, Quoc Hung. On the dynamics of the roots of polynomials under differentiation [J]. JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES,2022,162:1-22.
APA	Alazard, Thomas,Lazar, Omar,&Nguyen, Quoc Hung.(2022). On the dynamics of the roots of polynomials under differentiation .JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES,162,1-22.
MLA	Alazard, Thomas,et al." On the dynamics of the roots of polynomials under differentiation ".JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES 162(2022):1-22.