{"data":{"id":"10.13016/m25d0b","type":"dois","attributes":{"doi":"10.13016/m25d0b","prefix":"10.13016","suffix":"m25d0b","identifiers":[],"alternateIdentifiers":[],"creators":[{"name":"Markou, Ioannis","nameType":"Personal","givenName":"Ioannis","familyName":"Markou","affiliation":[],"nameIdentifiers":[]}],"titles":[{"title":"A Fokker-Planck Study Motivated by a Problem in Fluid-Particle Interactions"}],"publisher":"Digital Repository at the University of Maryland","container":{},"publicationYear":2014,"subjects":[{"subject":"Mathematics","subjectScheme":"pqcontrolled"},{"subject":"FOS: Mathematics","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"Applied mathematics","subjectScheme":"pqcontrolled"},{"subjectScheme":"pquncontrolled"}],"contributors":[],"dates":[{"date":"2014","dateType":"Issued"}],"language":"en","types":{"ris":"THES","bibtex":"phdthesis","citeproc":"thesis","schemaOrg":"Thesis","resourceType":"Dissertation","resourceTypeGeneral":"Collection"},"relatedIdentifiers":[],"relatedItems":[],"sizes":[],"formats":[],"version":null,"rightsList":[],"descriptions":[{"description":"This dissertation is a study of problems that relate to a Fokker-Planck (Klein-Kramers)\n\nequation with hypoelliptic structure. The equation describes the statistics of motion\n\nof an ensemble of particles in a viscous fluid that follows the Stokes ’ equations of\n\nfluid motion. The significance in this problem is that it relates to a variety of phenomena\n\nbesides its obvious connection to the study of macromolecular chains that are composed by\n\nparticle “ units ” in creeping flows. Such phenomena range from Kramers escape probability\n\n(for a particle trapped in a potential well), to stellar dynamics. The problem can also be\n\nseen as a simplified version of the Vlasov-Poisson-Fokker-Planck system that mainly describes\n\nelectrostatic models in plasma physics and gravitational forces between galaxies.\n\nWell-posedeness of the equation has been studied by many authors, including the\n\ncase of irregular coefficients (Lions-Le Bris). The study of Sobolev regularity\n\nis interesting in its own right and can be performed with fairly elementary tools (He\\'rau,Villani,…).\n\nWe are interested here with short time estimates and with how smoothing proceeds in time.\n\nDifferent types of Lyapunov functionals can be constructed depending on the\n\ntype of initial data to show regularization. Of particular interest is a recent\n\ntechnique developed by C.Villani that builds upon a system of differential inequalities\n\nand is being implemented here for the slightly more involved case of non constant friction.\n\nThe question of asymptotic convergence to a stationary state is also discussed, with\n\ntechniques that are similar to certain extend to the ones used in regularization but which in general\n\ninvolve more computations.\n\nFinally, we examine the hydrodynamic (zero mass) limit of the parametrized version of the\n\nFokker-Planck equation. We discuss two different approaches of hydrodynamic convergence.\n\nThe first uses weak compactness principles\n\nof extracting subsequences that are shown to converge to a solution of the limit problem, and\n\nworks with initial data in weighted L^{2} setting. The second is based\n\non the study of relative entropy, gives L^{1} convergence to a solution of the limit problem, and\n\nuses entropic initial data.","descriptionType":"Abstract"}],"geoLocations":[],"fundingReferences":[],"xml":"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","url":"http://hdl.handle.net/1903/16260","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-2.2","source":null,"isActive":true,"state":"findable","reason":null,"viewCount":0,"viewsOverTime":[],"downloadCount":0,"downloadsOverTime":[],"referenceCount":0,"citationCount":0,"citationsOverTime":[],"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2015-02-07T06:33:20.000Z","registered":"2015-02-07T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