Gelfand-Shilov and Gevrey smoothing effect for the spatially   inhomogeneous non-cutoff Kac equation
  We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. 
Yoshinori Morimoto, Nicolas Lerner, Karel Pravda-Starov, Chao-Jiang Xu
math.AP