{"title":"Boltzmann方程越过障碍物的Hölder正则性","authors":"Chanwoo Kim, Donghyun Lee","doi":"10.1002/cpa.22167","DOIUrl":null,"url":null,"abstract":"<p>Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in <math>\n <semantics>\n <msubsup>\n <mi>C</mi>\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mi>v</mi>\n </mrow>\n <mrow>\n <mn>0</mn>\n <mo>,</mo>\n <mfrac>\n <mn>1</mn>\n <mn>2</mn>\n </mfrac>\n <mo>−</mo>\n </mrow>\n </msubsup>\n <annotation>$C^{0,\\frac{1}{2}-}_{x,v}$</annotation>\n </semantics></math> for the Boltzmann equation of the hard-sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in-flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 4","pages":"2331-2386"},"PeriodicalIF":3.1000,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hölder regularity of the Boltzmann equation past an obstacle\",\"authors\":\"Chanwoo Kim, Donghyun Lee\",\"doi\":\"10.1002/cpa.22167\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in <math>\\n <semantics>\\n <msubsup>\\n <mi>C</mi>\\n <mrow>\\n <mi>x</mi>\\n <mo>,</mo>\\n <mi>v</mi>\\n </mrow>\\n <mrow>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mfrac>\\n <mn>1</mn>\\n <mn>2</mn>\\n </mfrac>\\n <mo>−</mo>\\n </mrow>\\n </msubsup>\\n <annotation>$C^{0,\\\\frac{1}{2}-}_{x,v}$</annotation>\\n </semantics></math> for the Boltzmann equation of the hard-sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in-flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].</p>\",\"PeriodicalId\":10601,\"journal\":{\"name\":\"Communications on Pure and Applied Mathematics\",\"volume\":\"77 4\",\"pages\":\"2331-2386\"},\"PeriodicalIF\":3.1000,\"publicationDate\":\"2023-10-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22167\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22167","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Hölder regularity of the Boltzmann equation past an obstacle
Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard-sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in-flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].
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