{"title":"Smoothing does not give a selection principle for transport equations with bounded autonomous fields","authors":"Camillo De Lellis, Vikram Giri","doi":"10.1007/s40316-021-00160-y","DOIUrl":null,"url":null,"abstract":"<div><p>We give an example of a bounded divergence free autonomous vector field in <span>\\({\\mathbb {R}}^3\\)</span> (and of a nonautonomous bounded divergence free vector field in <span>\\({\\mathbb {R}}^2\\)</span>) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.</p></div>","PeriodicalId":42753,"journal":{"name":"Annales Mathematiques du Quebec","volume":"46 1","pages":"27 - 39"},"PeriodicalIF":0.5000,"publicationDate":"2021-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40316-021-00160-y","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales Mathematiques du Quebec","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40316-021-00160-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
We give an example of a bounded divergence free autonomous vector field in \({\mathbb {R}}^3\) (and of a nonautonomous bounded divergence free vector field in \({\mathbb {R}}^2\)) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.
期刊介绍：
The goal of the Annales mathématiques du Québec (formerly: Annales des sciences mathématiques du Québec) is to be a high level journal publishing articles in all areas of pure mathematics, and sometimes in related fields such as applied mathematics, mathematical physics and computer science.
Papers written in French or English may be submitted to one of the editors, and each published paper will appear with a short abstract in both languages.
History:
The journal was founded in 1977 as „Annales des sciences mathématiques du Québec”, in 2013 it became a Springer journal under the name of “Annales mathématiques du Québec”. From 1977 to 2018, the editors-in-chief have respectively been S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea.
Les Annales mathématiques du Québec (anciennement, les Annales des sciences mathématiques du Québec) se veulent un journal de haut calibre publiant des travaux dans toutes les sphères des mathématiques pures, et parfois dans des domaines connexes tels les mathématiques appliquées, la physique mathématique et l''informatique.
On peut soumettre ses articles en français ou en anglais à l''éditeur de son choix, et les articles acceptés seront publiés avec un résumé court dans les deux langues.
Histoire:
La revue québécoise “Annales des sciences mathématiques du Québec” était fondée en 1977 et est devenue en 2013 une revue de Springer sous le nom Annales mathématiques du Québec. De 1977 à 2018, les éditeurs en chef ont respectivement été S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea.
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