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.

中文翻译：

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杂质的大气扩散问题的渐近稳定周期解：渐近性，存在性和唯一性

摘要

这项工作的基础是在反应扩散对流问题中使用现代渐近分析方法，以描述非线性扩散对流方程奇异摄动问题的经典边界层周期解。构造了这种解的任意阶的渐近逼近，并证明了形式构造是合理的。证明了唯一性定理，建立了渐近Lyapunov稳定性，并找到了边界层周期解吸引的局部域。讨论了该结果在大气扩散问题中的应用之一，即人为杂质在大气边界层中的迁移和化学转化过程的数学模型，并考虑了周期性（例如）每日或季节性变化。针对此问题开发的分析算法也将成为计算城市来源人为杂质每日校正排放通量的新方法的基础，这将有可能开发出改进的方法来确定某地区整个区域的每日总排放量。城市或城市群，基于模型问题的解析解决方案并结合在大气监测站网络上获得的信息。