This talk expands on the first result of the kinetic stability described in R. Esposito's talk. In the context of the Rayleigh-Benard problem the stability of two dimensional convective solutions to the stationary Boltzmann equation which are close to a roll solution near above the bifurcation point from laminar to convective solutions is proven.
First a survey of the steps to reach the result is performed: the expansion of the solution in tems of an asymptotic expansion plus a rest term, the nonnegativity of the solution, the use of an appropriate spectral estimate, the control of the hydrodynamic moments of the solution, the stability proof.
The control of the hydrodynamic moments of the solution is the most delicate piece of the proof. Hints on it are given in the second part of the talk.