{"title":"Gevrey regularity for the Vlasov-Poisson system","authors":"Renato Velozo Ruiz","doi":"10.1016/j.anihpc.2020.10.006","DOIUrl":null,"url":null,"abstract":"<div><p>We prove propagation of <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mfrac></math></span>-Gevrey regularity <span><math><mo>(</mo><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>)</mo></math></span> for the Vlasov-Poisson system on <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> using a Fourier space method in analogy to the results proved for the 2D-Euler system in <span>[20]</span> and <span>[23]</span>. More precisely, we give quantitative estimates for the growth of the <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mfrac></math></span>-Gevrey norm and decay of the regularity radius for the solution of the system in terms of the force field and the volume of the support in the velocity variable of the distribution of matter. As an application, we show global existence of <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mfrac></math></span>-Gevrey solutions (<span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>) for the Vlasov-Poisson system in <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Furthermore, the propagation of Gevrey regularity can be easily modified to obtain the same result in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. In particular, this implies global existence of analytic <span><math><mo>(</mo><mi>s</mi><mo>=</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></mfrac></math></span>-Gevrey solutions (<span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>) for the Vlasov-Poisson system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.006","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0294144920301116","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We prove propagation of -Gevrey regularity for the Vlasov-Poisson system on using a Fourier space method in analogy to the results proved for the 2D-Euler system in [20] and [23]. More precisely, we give quantitative estimates for the growth of the -Gevrey norm and decay of the regularity radius for the solution of the system in terms of the force field and the volume of the support in the velocity variable of the distribution of matter. As an application, we show global existence of -Gevrey solutions () for the Vlasov-Poisson system in . Furthermore, the propagation of Gevrey regularity can be easily modified to obtain the same result in . In particular, this implies global existence of analytic and -Gevrey solutions () for the Vlasov-Poisson system in .
期刊介绍:
The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.