Asymptotic Problems with Boundary Effect in Kinetic Theory

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Kinetic theory describes the dynamics of a large number of particles, such as flows of air particles passing an airfoil, or neutrons' collisions in a nuclear reactor. In a statistical manner, the kinetic description bridges the micro-scale modeling of motion of particles by Newtonian mechanics and the macro-scale modeling by continuum fluid mechanics. This research project aims to develop novel mathematical methods to quantitatively characterize these multi-scale models. The investigation focuses on the motion of rarefied gas, neutrons, electrons, and ions, in spatially bounded regions under the influence of the surrounding environment. Its applications range from high-tech fields like semi-conductors or nuclear fusion, to daily-life devices like water sprays or fluorescent lamps.Specifically, this project concentrates on the hydrodynamic limit of the Boltzmann equation or transport equation, i.e. how the solution varies asymptotically when a small parameter, either the Knudsen number or the Strouhal number, approaches zero. In bounded domains, kinetic boundary corrections (i.e. boundary layers), play a crucial role. The investigator intends to justify the validity of the asymptotic approximation in the presence of singular boundary layers. Moreover, the initial-boundary interactions and nonlinear effects are taken into consideration. To tackle the non-standard asymptotic expansions, the investigator seeks to develop general theories of geometric correction in boundary layers, rescaled remainder estimates, Milne regularity analysis, non-local energy methods, and boundary layers decomposition. These techniques can be further applied to Vlasov systems and magnetohydrodynamical (MHD) equations. Also, the rigorous derivations of fractional diffusion and stochastic diffusion are involved.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.