{"title":"Well/Ill-Posedness Bifurcation for the Boltzmann Equation with Constant Collision Kernel","authors":"Xuwen Chen, Justin Holmer","doi":"10.1007/s40818-024-00177-w","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the 3D Boltzmann equation with the constant collision kernel. We investigate the well/ill-posedness problem using the methods from nonlinear dispersive PDEs. We construct a family of special solutions, which are neither near equilibrium nor self-similar, to the equation, and prove that the well/ill-posedness threshold in <span>\\(H^{s}\\)</span> Sobolev space is exactly at regularity <span>\\(s=1\\)</span>, despite the fact that the equation is scale invariant at <span>\\( s=\\frac{1}{2}\\)</span>.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"10 2","pages":""},"PeriodicalIF":2.4000,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pde","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40818-024-00177-w","RegionNum":1,"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
We consider the 3D Boltzmann equation with the constant collision kernel. We investigate the well/ill-posedness problem using the methods from nonlinear dispersive PDEs. We construct a family of special solutions, which are neither near equilibrium nor self-similar, to the equation, and prove that the well/ill-posedness threshold in \(H^{s}\) Sobolev space is exactly at regularity \(s=1\), despite the fact that the equation is scale invariant at \( s=\frac{1}{2}\).