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In this paper we investigate the non-linear Vlasov-Fokker-Planck (VFP) equation, a both physically and mathematically interesting modification of Vlasov's equation, which describes a plasma in a thermal bath. We prove existence, uniqueness and representation results for steady states of the VFP equation both in the case of a mollified interaction potential and for the VFP-Poisson system. The uniqueness and representation results are of special interest since they distinguish special solutions of the Vlasov equation.where E,, stands for some external electric field, Ein = -grad Vin, and p = p(t, x) is the spatial density of the plasma at time t. So iff=f(t, x, v) denotes the distribution of the plasma in phase space, p is given by p(t, x) = s f ( t , x, v)dv.

We study stationary solutions of the Vlasov‐Fokker‐Planck Equation, a modification of the Vlasov‐Poisson Equation, obtained by adding a diffusion term with respect to velocity. From a physical point of view it describes a plasma in thermal equilibrium. We prove existence of stationary solutions; for the mollified equation even an existence‐ and uniqueness theorem holds for sufficiently high temperature.

In recent years so-called 'virtual test rigs' have become more and more important in the development process of cars and trucks. Originally, the idea was to substitute expensive durability tests with computer simulation. Meanwhile, the focus has changed towards a more cooperative usage of numerical and laboratory rig simulation. For many safety critical issues laboratory tests remain indispensable. In early development stages, when no physical prototypes are available yet, numerical simulation is used to analyse and optimise the design. In this paper, we show how to build numerical simulation models of complex servo-hydraulic test systems and their test specimen using multi-body simulation for the mechanics in combination with simulation models for the hydraulics and controls. We illustrate this at two industrial application examples: a spindle-coupled passenger car suspension rig and a tyre-coupled full vehicle rig. We show how the simulation models are used to design and optimise better test rigs and to support the test rig operation by preparing the physical tests with new specimen, i.e. by performing numerical simulations including numerical drive file iteration before the physical tests start