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dc.contributor.advisorLevermore, Daviden_US
dc.contributor.authorCampini, Marco.
dc.creatorCampini, Marco.en_US
dc.date.accessioned2011-10-31T17:44:43Z
dc.date.available2011-10-31T17:44:43Z
dc.date.issued1991en_US
dc.identifier.urihttp://hdl.handle.net/10150/185664
dc.description.abstractThe old question concerning the mathematical formulation of the fluid dynamic limits of kinetic theory is examined by studying the solution of the Cauchy problem for two differently scaled linearized Boltzmann equations on periodic domain as the mean free path of the particles becomes small. Under minimal assumptions on the initial data, by using an a priori estimate, it is possible, in a Hilbert space functional frame, to prove the weak convergence of solutions toward a function that has the form of an infinitesimal maxwellian in the velocity variable. The velocity moments of this function are then proved to satisfy either the linearized Euler or the Stokes system of equations (depending on the chosen scaling), by passing to the limit in the conservation relations derived from the Boltzmann equation. A theorem injecting continuously the intersection of certain weak spaces into a normed one is proved. Together with properties of the Euler semigroup, this allows to show strong convergence of the first three moments of the distribution function toward the macroscopic quantities density, bulk velocity and temperature, solutions of the linearized Euler system. The Stokes case is treated somewhat differently, through the introduction of a result, proved by using the adjoint formulation for linear kinetic equations, that extends the averaging theory of Golse-Lions-Perthame-Sentis. The desired convergence for the divergence-free component of the second moment toward the macroscopic velocity is then shown.
dc.language.isoenen_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.subjectDissertations, Academicen_US
dc.subjectMathematicsen_US
dc.subjectFluid dynamicsen_US
dc.subjectTransport theory.en_US
dc.titleThe fluid dynamical limits of the linearized Boltzmann equation.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.identifier.oclc711880859en_US
thesis.degree.grantorUniversity of Arizonaen_US
thesis.degree.leveldoctoralen_US
dc.contributor.committeememberFlaschka, Hermannen_US
dc.contributor.committeememberBayly, Bruceen_US
dc.identifier.proquest9210272en_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.namePh.D.en_US
refterms.dateFOA2018-06-12T11:48:01Z
html.description.abstractThe old question concerning the mathematical formulation of the fluid dynamic limits of kinetic theory is examined by studying the solution of the Cauchy problem for two differently scaled linearized Boltzmann equations on periodic domain as the mean free path of the particles becomes small. Under minimal assumptions on the initial data, by using an a priori estimate, it is possible, in a Hilbert space functional frame, to prove the weak convergence of solutions toward a function that has the form of an infinitesimal maxwellian in the velocity variable. The velocity moments of this function are then proved to satisfy either the linearized Euler or the Stokes system of equations (depending on the chosen scaling), by passing to the limit in the conservation relations derived from the Boltzmann equation. A theorem injecting continuously the intersection of certain weak spaces into a normed one is proved. Together with properties of the Euler semigroup, this allows to show strong convergence of the first three moments of the distribution function toward the macroscopic quantities density, bulk velocity and temperature, solutions of the linearized Euler system. The Stokes case is treated somewhat differently, through the introduction of a result, proved by using the adjoint formulation for linear kinetic equations, that extends the averaging theory of Golse-Lions-Perthame-Sentis. The desired convergence for the divergence-free component of the second moment toward the macroscopic velocity is then shown.


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