{"title":"Carathéodory theory and a priori estimates for continuity inclusions in the space of probability measures","authors":"Benoît Bonnet-Weill , Hélène Frankowska","doi":"10.1016/j.na.2024.113595","DOIUrl":null,"url":null,"abstract":"<div><p>In this article, we extend the foundations of the theory of differential inclusions in the space of compactly supported probability measures, introduced recently by the authors, to the setting of general Wasserstein spaces. In this context, we prove a novel existence result à la Peano for this class of dynamics under mere Carathéodory regularity assumptions. The latter is based on a natural, yet previously unexplored set-valued adaptation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations whose right-hand sides are measurable in the time variable. By leveraging some of the underlying methods along with new estimates for solutions of continuity equations, we also bring substantial improvements to the earlier versions of the Filippov estimates, compactness and relaxation properties of the solution sets of continuity inclusions, which are derived in the Cauchy–Lipschitz framework.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"247 ","pages":"Article 113595"},"PeriodicalIF":1.3000,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24001147","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we extend the foundations of the theory of differential inclusions in the space of compactly supported probability measures, introduced recently by the authors, to the setting of general Wasserstein spaces. In this context, we prove a novel existence result à la Peano for this class of dynamics under mere Carathéodory regularity assumptions. The latter is based on a natural, yet previously unexplored set-valued adaptation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations whose right-hand sides are measurable in the time variable. By leveraging some of the underlying methods along with new estimates for solutions of continuity equations, we also bring substantial improvements to the earlier versions of the Filippov estimates, compactness and relaxation properties of the solution sets of continuity inclusions, which are derived in the Cauchy–Lipschitz framework.
在本文中,我们将作者最近提出的紧凑支撑概率量空间中微分夹杂理论的基础扩展到一般瓦瑟斯坦空间的环境中。在此背景下,我们证明了在单纯的卡拉瑟奥多里正则性假设下这一类动力学的新颖存在性结果(à la Peano)。后者基于菲利波夫提出的半离散欧拉方案的一个自然的、但以前从未探索过的集合值适应性,以研究其右手边在时间变量中可测量的常微分方程。通过利用一些基础方法和对连续性方程解的新估计,我们还对早期版本的菲利波夫估计、连续性夹杂解集的紧凑性和松弛特性进行了实质性改进,这些都是在 Cauchy-Lipschitz 框架中推导出来的。
期刊介绍:
Nonlinear Analysis focuses on papers that address significant problems in Nonlinear Analysis that have a sustainable and important impact on the development of new directions in the theory as well as potential applications. Review articles on important topics in Nonlinear Analysis are welcome as well. In particular, only papers within the areas of specialization of the Editorial Board Members will be considered. Authors are encouraged to check the areas of expertise of the Editorial Board in order to decide whether or not their papers are appropriate for this journal. The journal aims to apply very high standards in accepting papers for publication.