Classical stability-bifurcation scenarios in Hydrodynamics and Thermodynamics can in some cases be analyzed on the kinetic level. Two examples where such kinetic analysis is possible are presented. The first one is the Benard problem of a fluid heated from below. In this case the purely conductive stationary solution looses its stability when the Rayleigh number crosses a critical value and stable convective motion arise. The stability properties at hydrodynamic level are not expected to persist for arbitrarily large Knudsen numbers. It is proven instead that this is the case for small Knudsen numbers. More on this problem will follow in Anne Nouri's talk The second example is a binary interacting gas undergoing a segregation phase transtion. In this case the equilibrium state corresponding to the mixed phase, which is a minimizer for the free energy at high temperature, becomes a local maximizer when the temperature goes below a critical value and new minimizers appear, corresponding to segregation of the two species in different regions.
On the kinetic level this is modeled by a self-consistent force between different species which are driven to equilibrium by Boltzmann collisions. In this case it is possible to show, for arbitrary Knudsen numbers, the stability of the mixed phase at high temperature and of the segregated equilibrium at low temperature, where the mixed phase is instead proved to be unstable.
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