Abstract:
In this talk, a general framework to study general class of linear and nonlinear kinetic equations with random uncertainties from the initial data and/or collision kernels, and their stochastic Galerkin approximationsâ€“in both incompressible Navier-Stokes and Euler (acoustic) regimesâ€“is provided. We show that the general framework put forth in [C. Mouhot and L. Neumann, Nonlinearity, 19, 969-998, 2006; M. Briant, J. Di. Eqn., 259, 6072-6141, 2005] based on hypocoercivity for the deterministic kinetic equations can be 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. An extension of the theory to the bipolar semiconductor Boltzmann system with random inputs and small scalings will also be briefly discussed.
This is a joint work with Shi Jin. |