Asymptotically Stable Periodic Solutions in One Problem of Atmospheric Diffusion of Impurities: Asymptotics, Existence, and Uniqueness

Abstract

The basis of this work is the use of modern methods of asymptotic analysis in reaction–diffusion–advection problems in order to describe the classical boundary-layer periodic solution of one singularly perturbed problem for the nonlinear diffusion–advection equation. An asymptotic approximation of an arbitrary order of such a solution is constructed, and the formal construction is justified. The uniqueness theorem is proved, the asymptotic Lyapunov stability is established, and the local domain of attraction of the boundary-layer periodic solution is found. One of the applications of this result to atmospheric diffusion problems is discussed, namely, mathematical modeling of the processes of transport and chemical transformation of anthropogenic impurities in the atmospheric boundary layer with allowance for periodic, e.g., daily or seasonal changes. The analytical algorithms developed for this problem as well will form the basis for a new method for calculating daily corrected emission fluxes of anthropogenic impurities from urban sources, which will make it possible to develop improved methods for determining daily integral emissions from the entire territory of a city or a urban agglomeration, based on the use of analytical solutions of model problems in combination with information obtained on a network of atmospheric monitoring stations.