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This paper is concerned with a kineitc-fluid model with random initial inputsin the fine particle regime, which is a system coupling the incompressibleNavier-Stokes equations and the Vlasov-Fokker-Planck equations that modeldispersed particles of different sizes. A uniform regularity for random initialdata near the global equilibrium is established in some suitable Sobolev spacesby using energy estimates, and we also prove the energy decays exponentially intime by hypocoercivity arguments, which means that the long time behavior ofthe solution is insensitive to the random perturbation in the initial data. Forthe generalized polynomial chaos stochastic Galerkin method (gPC-sG) for themodel, with initial data near the global equilibrium and smooth enough in thephysical and random spaces, we prove that the gPC-sG method has spectralaccuracy, uniformly in time and the Knudsen number, and the error decaysexponentially in time.

Energy estimates and hypocoercivity analysis for a multi-phase Navier-Stokes-Vlasov-Fokker-Planck system with uncertainty