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We complete the result in  by showing the exponential decay of the perturbation of the laminar solution below the critical Rayleigh number and of the convective solutions above the critical Rayleigh number, in the kinetic framework.
We study a rarefied gas, described by the Boltzmann equation, between two coaxial rotating cylinders in the small Knudsen number regime. When the radius of the inner cylinder is suitably sent to infinity, the limiting evolution is expected to converge to a modified Couette flow which keeps memory of the vanishing curvature of the cylinders ( ghost effect ). In the $1$-d stationary case we prove the existence of a positive isolated $L_2$-solution to the Boltzmann equation and its convergence. This is obtained by means of a truncated bulk-boundary layer expansion which requires the study of a new Milne problem, and an estimate of the remainder based on a generalized spectral inequality.
We establish local-in-time validity of the Boltzmann equation in the presence of an external force deriving from a $C^2$ potential.
We consider a kinetic model for a system of two species of particles on a sufficiently large periodic interval, interacting through a long range repulsive potential and by collisions. The model is described by a set of two coupled Vlasov-Boltzmann equations. For temperatures below the critical value and suitably prescribed masses, there is a non homogeneous solution, the double soliton, which is a minimizer of the entropy functional. We prove the stability, up to translations, of the double soliton under small perturbations. The same arguments imply the stability of the pure phases, as well as the stability of the mixed phase above the critical temperature. The mixed phase is proved to be unstable below the critical temperature.
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