We will present the recent work by Tai-Ping Liu and the author on the stable, unstable and center manifolds for the stationary Boltzmann equation. The main technique is the time-asymptotic analysis based on the Green's function. Among the applications, we give an analytical understanding of the Sone's diagram of bifurcation phenomena for complete condensation boundary problem. We also show that the Boltzmann shock profiles are monotone.
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