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.
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REFERENCES
N. N. Nefedov, “The method of differential inequalities for some singularly perturbed partial differential equations,” Differ. Equations 31 (4), 668–671 (1995).
N. N. Nefedov, “An asymptotic method of differential inequalities for the investigation of periodic contrast structures: Existence, asymptotics, and stability,” Differ. Equations 36 (2), 298–305 (2000).
N. N. Nefedov and M. A. Davydova, “Periodic contrast structures in systems of reaction–diffusion–advection type,” Differ. Equations 46 (9), 688–706 (2010).
A. B. Vasil’eva and M. A. Davydova, “On a contrast steplike structure for a class of second-order nonlinear singularly perturbed equations,” Comput. Math. Math. Phys. 38 (6), 900–908 (1998).
N. N. Nefedov and E. I. Nikulin, “Existence and stability of periodic contrast structures in the reaction–advection–diffusion problem,” Russ. J. Math. Phys. 22 (2), 215–226 (2015).
N. N. Nefedov, E. I. Nikulin, and L. Recke, “On the existence and asymptotic stability of periodic contrast structures in quasilinear reaction–advection–diffusion equations,” Russ. J. Math. Phys. 26 (1), 55–69 (2019).
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations (Vysshaya Shkola, Moscow, 1990) [in Russian].
A. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, “Singularly perturbed problems with boundary and internal layers,” Proc. Steklov Inst. Math. 268, 258–273 (2010).
H. Amann, “Periodic solutions of semilinear parabolic equations,” in Nonlinear Analysis: Collections of Papers in Honor of Erich Rothe (Academic, New York, 1978), pp. 1–29.
P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series (Longman Scientific and Technical, New York, 1991), pp. 62–80.
N. F. Elansky, O. V. Lavrova, A. I. Skorokhod, and I. B. Belikov, “Trace gases in the atmosphere over Russian cities,” Atm. Environ. 143, 108–119 (2016).
B. R. Gurjar, T. M. Butler, M. G. Lawrence, and J. Lelieveld, “Evaluation of emissions and air qualities in megacities,” Atm. Environ. 42, 1593–1606 (2008).
N. Elansky, “Air quality and CO emissions in the Moscow megacity,” Urban Clim. 8, 42–56 (2014).
M. E. Berlyand, Prediction and Control of Atmospheric Pollution (Gidrometeoizdat, Leningrad, 1985).
M. A. Davydova, “Existence and stability of solutions with boundary layers in multidimensional singularly perturbed reaction-diffusion-advection problems,” Math. Notes 98, 909–919 (2015).
ACKNOWLEDGMENTS
The authors are thankful to Prof. V.F. Butuzov for the discussion of the results of this work and useful recommendations for improving it.
Funding
This work was supported by the Russian Foundation for Basic Research, project no. 18-29-10080.
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Translated by E. Chernokozhin
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Davydova, M.A., Nechaeva, A.L. Asymptotically Stable Periodic Solutions in One Problem of Atmospheric Diffusion of Impurities: Asymptotics, Existence, and Uniqueness. Comput. Math. and Math. Phys. 60, 448–458 (2020). https://doi.org/10.1134/S0965542520030070
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DOI: https://doi.org/10.1134/S0965542520030070