Kiyoshi Mizohata

The global weak solutions of compressible Euler equation with spherical symmetry

We shall study the compressible Euler equation which describes the motion of an isentropic gas. Many global existence theorems have been obtained for the one dimensional case. On the other hand, little is known for the casen>-2. No global weak solutions have been known to exist, but only local classical solutions. In this paper, we will present global weak solutions first for the casen>-2. We will do this, however, only for the case of spherical symmetry with γ=1, by using a modified Glimm’s method.

Global weak solutions of the compressible Euler equation with spherical symmetry (II)

We study the spherically symmetric motion of isothermal gas surrounding a ball. Recently we constructed global weak solutions for the initial boundary value problem in [2], but the class of the initial data considered there was not wide enough to include the stationary solutions for which the density is constant on the whole space. In this paper we will extend our preceding result to a wider class of the initial data so that it includes such stationary solutions. To do this, we will present a modification of Glimm’s method in which the mesh lengths for approximate solutions are not uniform.

Global solutions to the relativistic Euler equation with spherical symmetry

AbstractThe relativistic Euler equation inR3 is given by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB% PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x% c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr% Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaam% aalaaabaGaeyOaIylabaGaeyOaIy7exLMBb50ujbqeguuDJXwAKbac% fiGae8hDaqhaaGqaaiaa+bcadaqadaqaamaalaaabaGaeqyWdiNaam% 4yamaaCaaaleqabaacfaGae0NmaidaaOGaey4kaSIaamiuaaqaaiaa% dogadaahaaWcbeqaaiab9jdaYaaaaaGccaGFGaWaaSaaaeaacaGFXa% aabaGaa4xmaGGaaiab8jHiTmaalaaabaqeduuDJXwAKbYu51MyVXga% iyGacqWEfpqDdaahaaWcbeqaaiab9jdaYaaaaOqaaiab-ngaJnaaCa% aaleqabaGae0NmaidaaaaaaaGccqGHsisldaWcaaqaaiaadcfaaeaa% caWGJbWaaWbaaSqabeaacqqFYaGmaaaaaaGccaGLOaGaayzkaaGaa4% hiaiabgUcaRmaaqahabaWaaSaaaeaacqGHciITaeaacqGHciITcqWF% 4baEdaWgaaWcbaGae8NAaOgabeaaaaaabaGae8NAaOMaeWxpa0Jaa4% xmaaqaaiab9ndaZaqdcqGHris5aOGaa4hiamaabmaabaWaaSaaaeaa% cqaHbpGCcaWGJbWaaWbaaSqabeaacqqFYaGmaaGccqGHRaWkcaWGqb% aabaGaam4yamaaCaaaleqabaGae0Nmaidaaaaakiaa+bcadaWcaaqa% aiab7v8a1naaBaaaleaacqWFQbGAaeqaaaGcbaGaa4xmaiab8jHiTm% aalaaabaGaeSxXdu3aaWbaaSqabeaacqqFYaGmaaaakeaacqWFJbWy% daahaaWcbeqaaiab9jdaYaaaaaaaaaGccaGLOaGaayzkaaGaa4hiai% aa+1dacaGFGaGae0hmaaJae0hlaWcabaWaaSaaaeaacqGHciITaeaa% cqGHciITcqWF0baDaaGaa4hiamaabmaabaWaaSaaaeaacqaHbpGCca% WGJbWaaWbaaSqabeaacqqFYaGmaaGccqGHRaWkcaWGqbaabaGaam4y% amaaCaaaleqabaGae0Nmaidaaaaakiaa+bcadaWcaaqaaiab7v8a1n% aaBaaaleaacqWFPbqAaeqaaaGcbaGaa4xmaiab8jHiTmaalaaabaGa% eSxXdu3aaWbaaSqabeaacqqFYaGmaaaakeaacqWFJbWydaahaaWcbe% qaaiab9jdaYaaaaaaaaaGccaGLOaGaayzkaaGaa4hiaiabgUcaRmaa% qahabaWaaSaaaeaacqGHciITaeaacqGHciITcqWF4baEdaWgaaWcba% Gae8NAaOgabeaaaaaabaGae8NAaOMaeWxpa0Jaa4xmaaqaaiab9nda% ZaqdcqGHris5aOGaa4hiamaabmaabaWaaSaaaeaacqaHbpGCcaWGJb% WaaWbaaSqabeaacqqFYaGmaaGccqGHRaWkcaWGqbaabaGaam4yamaa% CaaaleqabaGae0Nmaidaaaaakiaa+bcadaWcaaqaaiab7v8a1naaBa% aaleaacqWFPbqAaeqaaOGaeSxXdu3aaSbaaSqaaiab-PgaQbqabaaa% keaacaGFXaGaeWNeI0YaaSaaaeaacqWEfpqDdaahaaWcbeqaaiab9j% daYaaaaOqaaiab-ngaJnaaCaaaleqabaGae0NmaidaaaaaaaGccaGF% GaGaey4kaSIaa4hiaiabes7aKnaaBaaaleaacqWFPbqAcqWFQbGAae% qaaOGaamiuaaGaayjkaiaawMcaaiaa+bcacaGF9aGaa4hiaiab9bda% Wiab9XcaSiab9bcaGiab-LgaPjab-bcaGiab81da9iab-bcaGiab9f% daXiab9XcaSiab9jdaYiab9XcaSiab9ndaZiab-5caUaaaaa!D554!

The Analysis of Big Data by Wavelets

\begin{gathered} \frac{\partial }{{\partial t}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{1}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}} - \frac{P}{{c^2 }}} \right) + \sum\limits_{j = 1}^3 {\frac{\partial }{{\partial x_j }}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{{\upsilon _j }}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}} \right) = 0, \hfill \\ \frac{\partial }{{\partial t}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{{\upsilon _i }}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}} \right) + \sum\limits_{j = 1}^3 {\frac{\partial }{{\partial x_j }}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{{\upsilon _i \upsilon _j }}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}} + \delta _{ij} P} \right) = 0, i = 1,2,3. \hfill \\ \end{gathered}

Fluorescence observation supporting capillary chromatography based on tube radial distribution of carrier solvents under laminar flow conditions

. In 1993, Smoller and Temple have constructed global weak solutions to this equation for 1 dimensional case. In this article we succeed, to show the existence of global weak solutions with spherical symmetry. Suppose that solutions are spherically symmetric. Then the equation becomes % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB% PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x% c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr% Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaam% aalaaabaGaeyOaIylabaGaeyOaIy7exLMBb50ujbqeguuDJXwAKbac% fiGae8hDaqhaaGqaaiaa+bcadaqadaqaamaalaaabaGaeqyWdiNaam% 4yamaaCaaaleqabaacfaGae0NmaidaaOGaey4kaSIaamiuaaqaaiaa% dogadaahaaWcbeqaaiab9jdaYaaaaaGccaGFGaWaaSaaaeaacaGFXa% aabaGaa4xmaGGaaiab8jHiTmaalaaabaqeduuDJXwAKbYu51MyVXga% iyGacqWEfpqDdaahaaWcbeqaaiab9jdaYaaaaOqaaiab-ngaJnaaCa% aaleqabaGae0NmaidaaaaaaaGccqGHsisldaWcaaqaaiaadcfaaeaa% caWGJbWaaWbaaSqabeaacqqFYaGmaaaaaaGccaGLOaGaayzkaaGaa4% hiaiabgUcaRmaalaaabaGaeyOaIylabaGaeyOaIyRae8NCaihaaiaa% +bcadaqadaqaamaalaaabaGaeqyWdiNaam4yamaaCaaaleqabaGae0% NmaidaaOGaey4kaSIaamiuaaqaaiaadogadaahaaWcbeqaaiab9jda% YaaaaaGccaGFGaWaaSaaaeaacqWEfpqDaeaacaGFXaGaeWNeI0YaaS% aaaeaacqWEfpqDdaahaaWcbeqaaiab9jdaYaaaaOqaaiab-ngaJnaa% CaaaleqabaGae0NmaidaaaaaaaaakiaawIcacaGLPaaacaGFGaGaeW% 3kaSIaeWhiaaYaaSaaaeaacqaHbpGCcaWGJbWaaWbaaSqabeaacqqF% YaGmaaGccqGHRaWkcaWGqbaabaGaam4yamaaCaaaleqabaGae0Nmai% daaaaakiaa+bcadaWcaaqaaiaa+fdaaeaacaGFXaGaeWNeI0YaaSaa% aeaacqWEfpqDdaahaaWcbeqaaiab9jdaYaaaaOqaaiab-ngaJnaaCa% aaleqabaGae0NmaidaaaaaaaGcdaWcaaqaaiaaikdacqWEfpqDaeaa% cqWFYbGCaaGaa4hiaiabg2da9iaa+bcacqqFWaamcqqFSaalaeaada% WcaaqaaiabgkGi2cqaaiabgkGi2kab-rha0baacaGFGaWaaeWaaeaa% daWcaaqaaiabeg8aYjaadogadaahaaWcbeqaaiab9jdaYaaakiabgU% caRiaadcfaaeaacaWGJbWaaWbaaSqabeaacqqFYaGmaaaaaOGaa4hi% amaalaaabaGaeSxXduhabaGaa4xmaiab8jHiTmaalaaabaGaeSxXdu% 3aaWbaaSqabeaacqqFYaGmaaaakeaacqWFJbWydaahaaWcbeqaaiab% 9jdaYaaaaaaaaaGccaGLOaGaayzkaaGaa4hiaiabgUcaRmaalaaaba% GaeyOaIylabaGaeyOaIyRae8NCaihaaiaa+bcadaqadaqaamaalaaa% baGaeqyWdiNaam4yamaaCaaaleqabaGae0NmaidaaOGaey4kaSIaam% iuaaqaaiaadogadaahaaWcbeqaaiab9jdaYaaaaaGccaGFGaWaaSaa% aeaacqWEfpqDdaahaaWcbeqaaiab9jdaYaaaaOqaaiaa+fdacqaFsi% sldaWcaaqaaiab7v8a1naaCaaaleqabaGae0NmaidaaaGcbaGae83y% am2aaWbaaSqabeaacqqFYaGmaaaaaaaakiaa+bcacqGHRaWkcaGFGa% GaamiuaaGaayjkaiaawMcaaiaa+bcacqaFRaWkcqaFGaaidaWcaaqa% aiabeg8aYjaadogadaahaaWcbeqaaiab9jdaYaaakiabgUcaRiaadc% faaeaacaWGJbWaaWbaaSqabeaacqqFYaGmaaaaaOGaa4hiamaalaaa% baGaeSxXduhabaGaa4xmaiab8jHiTmaalaaabaGaeSxXdu3aaWbaaS% qabeaacqqFYaGmaaaakeaacqWFJbWydaahaaWcbeqaaiab9jdaYaaa% aaaaaOWaaSaaaeaacaaIYaGaeSxXdu3aaWbaaSqabeaacqqFYaGmaa% aakeaacqWFYbGCaaGaa4hiaiabg2da9iaa+bcacqqFWaamcqWFUaGl% aaaa!DF5A!

Equivalence of Eulerian and Lagrangian weak solutions of the compressible Euler equation with spherical symmetry

\begin{gathered} \frac{\partial }{{\partial t}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{1}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}} - \frac{P}{{c^2 }}} \right) + \frac{\partial }{{\partial r}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{\upsilon }{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}} \right) + \frac{{\rho c^2 + P}}{{c^2 }} \frac{1}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}\frac{{2\upsilon }}{r} = 0, \hfill \\ \frac{\partial }{{\partial t}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{\upsilon }{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}} \right) + \frac{\partial }{{\partial r}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{{\upsilon ^2 }}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}} + P} \right) + \frac{{\rho c^2 + P}}{{c^2 }} \frac{\upsilon }{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}\frac{{2\upsilon ^2 }}{r} = 0. \hfill \\ \end{gathered}