HYPOCOERCIVITY BASED SENSITIVITY ANALYSIS AND SPECTRAL CONVERGENCE OF THE STOCHASTIC GALERKIN APPROXIMATION TO COLLISIONAL KINETIC EQUATIONS WITH MULTIPLE SCALES AND RANDOM INPUTS

摘要

In this paper we provide a general framework to study a general class of linear and nonlinear kinetic equations with random uncertainties from the initial data or collision kernels, and their stochastic Galerkin (SG) 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 (2006), pp. 969-998; M. Briant, J. Differential Equations, 259 (2005), pp. 6072-6141] 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 SG methods for the equations, establish hypocoercivity of the SG system and regularity of its solution, and study spectral accuracy and exponential decay in time of the numerical error of the method in a weighted Sobolev norm.