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 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2021, Volume 192, Pages 84–93 (Mi into784)

On the asymptotics of the solution to the Cauchy problem for a singularly perturbed system of transfer equations with low nonlinear diffusion

A. V. Nesterov

Plekhanov Russian State University of Economics, Moscow

Abstract: This paper is a survey of results concerning asymptotics of solutions of singularly perturbed systems of transport equations; it also contains some new results. We discuss the so-called critical problems whose degenerate solutions are one-parameter families. Under certain conditions, this leads to a fast establishment of dynamic equilibrium between the components of the solution and the subsequent transfer with an “average” rate. The regions of large gradients of the initial conditions generate inner layers, which can be described by linear parabolic equations and their generalizations, for example, equations of the Burgers and Burgers–Korteweg–de Vries types.

Keywords: system of transport equations, singular perturbation, asymptotic expansion in a small parameter, critical case, parabolic transition layer, Burgers–Korteweg–de Vries equation

 Funding Agency This work was supported by a grant from the G. V. Plekhanov Russian University of Economics on the topic “Intelligent system for analyzing satellite data for predicting economic consequences of the dynamics of the global distribution of drinking water supplies and fire hazard.”

DOI: https://doi.org/10.36535/0233-6723-2021-192-84-93

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UDC: 517.955.8
MSC: 35F40, 35F55, 35F61, 35F99

Citation: A. V. Nesterov, “On the asymptotics of the solution to the Cauchy problem for a singularly perturbed system of transfer equations with low nonlinear diffusion”, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 3, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 192, VINITI, Moscow, 2021, 84–93

Citation in format AMSBIB
\Bibitem{Nes21} \by A.~V.~Nesterov \paper On the asymptotics of the solution to the Cauchy problem for a singularly perturbed system of transfer equations with low nonlinear diffusion \inbook Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 3 \serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. \yr 2021 \vol 192 \pages 84--93 \publ VINITI \publaddr Moscow \mathnet{http://mi.mathnet.ru/into784} \crossref{https://doi.org/10.36535/0233-6723-2021-192-84-93}