The 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:707790 |
| Date | January 2017 |
| Creators | Dietert, Helge |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.repository.cam.ac.uk/handle/1810/263044 |
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