{"title":"热、磁化、相对论弗拉索夫麦克斯韦系统经典解的均匀寿命","authors":"C. Cheverry, S. Ibrahim, Dayton Preissl","doi":"10.3934/krm.2021042","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function <inline-formula><tex-math id=\"M1\">\\begin{document}$ f(t,\\cdot) $\\end{document}</tex-math></inline-formula>. Magnetically confined plasmas are characterized by the presence of a strong <i>external</i> magnetic field <inline-formula><tex-math id=\"M2\">\\begin{document}$ x \\mapsto \\epsilon^{-1} \\mathbf{B}_e(x) $\\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\epsilon $\\end{document}</tex-math></inline-formula> is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent <i>internal</i> electromagnetic fields <inline-formula><tex-math id=\"M4\">\\begin{document}$ (E,B) $\\end{document}</tex-math></inline-formula> are supposed to be small. In the non-magnetized setting, local <inline-formula><tex-math id=\"M5\">\\begin{document}$ C^1 $\\end{document}</tex-math></inline-formula>-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\epsilon^{-1} $\\end{document}</tex-math></inline-formula> is large, standard results predict that the lifetime <inline-formula><tex-math id=\"M7\">\\begin{document}$ T_\\epsilon $\\end{document}</tex-math></inline-formula> of solutions may shrink to zero when <inline-formula><tex-math id=\"M8\">\\begin{document}$ \\epsilon $\\end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id=\"M9\">\\begin{document}$ 0 $\\end{document}</tex-math></inline-formula>. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (<inline-formula><tex-math id=\"M10\">\\begin{document}$ 0<T<T_\\epsilon $\\end{document}</tex-math></inline-formula>) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows <inline-formula><tex-math id=\"M11\">\\begin{document}$ f $\\end{document}</tex-math></inline-formula> remains at a distance <inline-formula><tex-math id=\"M12\">\\begin{document}$ \\epsilon $\\end{document}</tex-math></inline-formula> from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.</p>","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"24 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2021-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system\",\"authors\":\"C. Cheverry, S. Ibrahim, Dayton Preissl\",\"doi\":\"10.3934/krm.2021042\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function <inline-formula><tex-math id=\\\"M1\\\">\\\\begin{document}$ f(t,\\\\cdot) $\\\\end{document}</tex-math></inline-formula>. Magnetically confined plasmas are characterized by the presence of a strong <i>external</i> magnetic field <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ x \\\\mapsto \\\\epsilon^{-1} \\\\mathbf{B}_e(x) $\\\\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ \\\\epsilon $\\\\end{document}</tex-math></inline-formula> is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent <i>internal</i> electromagnetic fields <inline-formula><tex-math id=\\\"M4\\\">\\\\begin{document}$ (E,B) $\\\\end{document}</tex-math></inline-formula> are supposed to be small. In the non-magnetized setting, local <inline-formula><tex-math id=\\\"M5\\\">\\\\begin{document}$ C^1 $\\\\end{document}</tex-math></inline-formula>-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since <inline-formula><tex-math id=\\\"M6\\\">\\\\begin{document}$ \\\\epsilon^{-1} $\\\\end{document}</tex-math></inline-formula> is large, standard results predict that the lifetime <inline-formula><tex-math id=\\\"M7\\\">\\\\begin{document}$ T_\\\\epsilon $\\\\end{document}</tex-math></inline-formula> of solutions may shrink to zero when <inline-formula><tex-math id=\\\"M8\\\">\\\\begin{document}$ \\\\epsilon $\\\\end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id=\\\"M9\\\">\\\\begin{document}$ 0 $\\\\end{document}</tex-math></inline-formula>. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (<inline-formula><tex-math id=\\\"M10\\\">\\\\begin{document}$ 0<T<T_\\\\epsilon $\\\\end{document}</tex-math></inline-formula>) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows <inline-formula><tex-math id=\\\"M11\\\">\\\\begin{document}$ f $\\\\end{document}</tex-math></inline-formula> remains at a distance <inline-formula><tex-math id=\\\"M12\\\">\\\\begin{document}$ \\\\epsilon $\\\\end{document}</tex-math></inline-formula> from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.</p>\",\"PeriodicalId\":49942,\"journal\":{\"name\":\"Kinetic and Related Models\",\"volume\":\"24 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2021-03-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Kinetic and Related Models\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3934/krm.2021042\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kinetic and Related Models","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/krm.2021042","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system
This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function \begin{document}$ f(t,\cdot) $\end{document}. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field \begin{document}$ x \mapsto \epsilon^{-1} \mathbf{B}_e(x) $\end{document}, where \begin{document}$ \epsilon $\end{document} is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields \begin{document}$ (E,B) $\end{document} are supposed to be small. In the non-magnetized setting, local \begin{document}$ C^1 $\end{document}-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since \begin{document}$ \epsilon^{-1} $\end{document} is large, standard results predict that the lifetime \begin{document}$ T_\epsilon $\end{document} of solutions may shrink to zero when \begin{document}$ \epsilon $\end{document} goes to \begin{document}$ 0 $\end{document}. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (\begin{document}$ 0) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows \begin{document}$ f $\end{document} remains at a distance \begin{document}$ \epsilon $\end{document} from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.
期刊介绍:
KRM publishes high quality papers of original research in the areas of kinetic equations spanning from mathematical theory to numerical analysis, simulations and modelling. It includes studies on models arising from physics, engineering, finance, biology, human and social sciences, together with their related fields such as fluid models, interacting particle systems and quantum systems. A more detailed indication of its scope is given by the subject interests of the members of the Board of Editors. Invited expository articles are also published from time to time.