Geodesic
On the asymptotics of the solution to the Cauchy problem for a singularly perturbed system of transfer equations with low nonlinear diffusion
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, 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, Tome 192 (2021), pp. 84-93.

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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, asymptotic expansion in a small parameter, critical case, parabolic transition layer, Burgers–Korteweg–de Vries equation.
Mots-clés : singular perturbation 
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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. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, 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, Tome 192 (2021), pp. 84-93. http://geodesic.mathdoc.fr/item/INTO_2021_192_a8/

[1] Vasileva A. B., “O vnutrennem perekhodnom sloe v reshenii sistemy uravnenii v chastnykh proizvodnykh pervogo poryadka”, Differ. uravn., 21:9 (1985), 1537–1544

[2] Vasileva A. B., Butuzov V. F., Singulyarno vozmuschennye uravneniya v kriticheskikh sluchayakh, Izd-vo MGU, M., 1978

[3] Nesterov A. V., “Ob asimptotike s perekhodnym sloem resheniya odnoi signgulyarno vozmuschennoi giperbolicheskoi sistemy uravnenii”, Dokl. AN SSSR., 305:6 (1989), 1350–1353

[4] Nesterov A. V., “Ob asimptotike resheniya singulyarno vozmuschennoi giperbolicheskoi sistemy uravnenii s maloi nelineinostyu v kriticheskom sluchae”, Zh. vychisl. mat. mat. fiz., 52:7 (2012), 1035–1043

[5] Nesterov A. V., “O strukture resheniya odnogo klassa giperbolicheskikh sistem s neskolkimi prostranstvennymi peremennymi v dalnei zone”, Zh. vychisl. mat. mat. fiz., 56:4 (2016), 639–649

[6] Nesterov A. V., “Asimptotika resheniya zadachi Koshi dlya singulyarno vozmuschennoi sistemy uravnenii perenosa s maloi diffuziei”, Sovremennye metody teorii kraevykh zadach, Mat. Mezhdunar. konf. «Voronezhskaya vesennyaya matematicheskaya shkola «Pontryaginskie chteniya–XXX» (3–9 maya 2019 g., Voronezh), Voronezh, 2019, 212–213

[7] Nesterov A. V., Shuliko O. V., “Asimptotika resheniya singulyarno vozmuschennoi sistemy differentsialnykh uravnenii pervogo poryadka v chastnykh proizvodnykh s maloi nelineinostyu v kriticheskom sluchae”, Zh. vychisl. mat. mat. fiz., 47:3 (2007), 438–444

[8] Nesterov A. V., Shuliko O. V., “Ob asimptotike resheniya singulyarno vozmuschennoi sistemy parabolicheskikh uravnenii v kriticheskom sluchae”, Zh. vychisl. mat. mat. fiz., 50:2 (2010), 268–275

[9] Nesterov A. V., Pavlyuk T. V., “Ob asimptotike resheniya singulyarno vozmuschennoi giperbolicheskoi sistemy s neskolkimi prostranstvennymi peremennymi v kriticheskom sluchae”, Zh. vychisl. mat. mat. fiz., 54:3 (2014), 450–462

[10] Rachinskii V. V., Vvedenie v obschuyu teoriyu dinamiki sorbtsii i khromatografii, Nauka, M., 1964

[11] Uizem Dzh., Lineinye i nelineinye volny, Mir, M., 1977