On the dynamics of the roots of polynomials under differentiation
文献类型：期刊论文
| 作者 | Alazard, Thomas2,3; Lazar, Omar4; Nguyen, Quoc Hung1 |
| 刊名 | JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
 |
| 出版日期 | 2022-06-01 |
| 卷号 | 162页码:1-22 |
| 关键词 | Transport equations Nonlocal operators Well-posedness Regularity |
| ISSN号 | 0021-7824 |
| DOI | 10.1016/j.matpur.2022.04.001 |
| 英文摘要 | 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. |
| 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 |
| 语种 | 英语 |
| WOS记录号 | WOS:000798180300001 |
| 出版者 | ELSEVIER |
| 源URL | [http://ir.amss.ac.cn/handle/2S8OKBNM/61433]  |
| 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.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China 2.Univ Paris Saclay, Ctr Borelli UMR9010, CNRS, ENS Paris-Saclay, Ave Sci, F-91190 Gif sur yvette, France 3.Univ Calif, Dept Math, Berkeley, CA 94720 USA 4.Amer Univ Middle East, Coll Engn & Technol, Egaila, Kuwait |
| 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. |
入库方式： OAI收割
来源：数学与系统科学研究院
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