  In this paper, we provide a general framework to study general class of
linear and nonlinear kinetic equations with random uncertainties from the
initial data or collision kernels, and their stochastic Galerkin
approximations, in both incompressible Navier-Stokes and Euler (acoustic)
regimes. First, we show that the general framework put forth in [C. Mouhot and
L. Neumann, Nonlinearity, 19, 969-998, 2006, M. Briant, J. Diff. Eqn., 259,
6072-6141, 2005] based on hypocoercivity for the deterministic kinetic
equations can be easily adopted for sensitivity analysis for random kinetic
equations, which gives rise to an exponential convergence of the random
solution toward the (deterministic) global equilibrium, under suitable
conditions on the collision kernel. Then we use such theory to study the
stochastic Galerkin (SG) methods for the equations, establish hypocoercivity of
the SG system and regularity of its solution, and spectral accuracy and
exponential decay of the numerical error of the method in a weighted Sobolev
norm.
