Uncertainty qualification of Vlasov-Poisson-Boltzmann equations in the diffusive scaling

Abstract

For the Vlasov-Poisson-Boltzmann equations with random uncertainties from the initial data or collision kernels, we proved the sensitivity analysis and energy estimates uniformly with respect to both the Knudsen number and the random variables in the diffusive scaling using hypocoercivity method developed in [6], [7], [14]. As a consequence, we also justified the incompressible Navier-Stokes-Poisson limit with random inputs. In particular, for the first time, we obtain the precise convergence rate without employing any results based on Hilbert expansion (in other words, we don't need any information from the limiting fluid equations a priori). We not only generalized the previous deterministic Navier-Stokes-Fourier-Poisson limits to random initial data case, but also improve the previous uncertainty quantification results to the case where the initial data include both kinetic and fluid parts. This is the first uncertainty qualification (UQ) result for spatially high dimension kinetic equations in diffusive limits containing Navier-Stokes dynamics, and generalizes the previous UQ results which does not contain fluid equations (for example, [34]).