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dc.contributor.authorDietert, Helge
dc.date.accessioned2017-03-13T11:10:01Z
dc.date.available2017-03-13T11:10:01Z
dc.date.issued2017-01-10
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/263044
dc.description.abstractThe main results of my work contribute to the mathematical study of a stability mechanism common to both the Vlasov–Poisson equation and the Kuramoto equation. These kinetic models come from very different areas of physics: the Vlasov–Poisson equation models plasmas and the Kuramoto equation models synchronisation behaviour. The stability was first described by Landau in 1946 and is a subtle behaviour, because the damping only happens in a suitably weak sense. In fact, the models are not dissipative and cannot be stable in a strong topology. Instead, the so-called Landau damping happens through phase mixing. My contributions include a simplified linear analysis for the Vlasov–Poisson equation around the spatially homogeneous state. For the Kuramoto equation, I cover the linear analysis around general stationary states and show nonlinear stability results with algebraic and exponential decay. Moreover, I show how the mean-field estimate by Dobrushin can be improved around the incoherent state. In addition, I study how a kinetic system can reach a thermal equilibrium. This is modelled by adding a dissipative term, which by itself drives the system to a local equilibrium. In hypocoercivity theory, the complementary effect of the transport operator is used to show exponential decay to a global equilibrium. In particular, I show how a probabilistic treatment can complement the standard hypocoercivity theory, which constructs equivalent norms, and I discuss the necessity of the geometric control condition for the spatially degenerate kinetic Fokker–Planck equation. Finally, I study the possible discretisation of the velocity variable for kinetic equations. For the numerical stability, Hermite functions are a suitable choice, because their differentiation matrix is skew-symmetric. However, so far a fast expansion algorithm has been lacking and this is addressed in this work.en
dc.language.isoenen
dc.publisherCambridge Centre for Analysis (CCA), Centre for Mathematical Sciences, University of Cambridgeen
dc.titleContributions to mixing and hypocoercivity in kinetic modelsen
dc.typeThesisen
dc.type.qualificationlevelDoctoral
dc.type.qualificationnameDoctor of Philosophy (PhD)
dc.publisher.institutionUniversity of Cambridgeen
dc.publisher.departmentDepartment of Applied Mathematics and Theoretical Physicsen
dc.publisher.departmentCambridge Centre for Analysis (CCA)en
dc.publisher.departmentQueens' Collegeen
dc.identifier.doi10.17863/CAM.7765
dc.contributor.orcidDietert, Helge [0000-0002-2071-9202]en
cam.supervisorMouhot, Clement
cam.supervisorIserles, Arieh


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