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JOURNALS // Uspekhi Matematicheskikh Nauk // Archive

Uspekhi Mat. Nauk, 2014 Volume 69, Issue 3(417), Pages 43–86 (Mi rm9584)

This article is cited in 10 papers

Boundary layer theory for convection-diffusion equations in a circle

Ch.-Y. Junga, R. Temamb

a School of Natural Science, Ulsan National Institute of Science and Technology, Ulsan, Republic of Korea
b The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, U.S.A.

Abstract: This paper is devoted to boundary layer theory for singularly perturbed convection-diffusion equations in the unit circle. Two characteristic points appear, $(\pm 1,0)$, in the context of the equations considered here, and singularities may occur at these points depending on the behaviour there of a given function $f$, namely, the flatness or compatibility of $f$ at these points as explained below. Two previous articles addressed two particular cases: [24] dealt with the case where the function $f$ is sufficiently flat at the characteristic points, the so-called compatible case; [25] dealt with a generic non-compatible case ($f$ polynomial). This survey article recalls the essential results from those papers, and continues with the general case ($f$ non-flat and non-polynomial) for which new specific boundary layer functions of parabolic type are introduced in addition.
Bibliography: 49 titles.

Keywords: boundary layers, singular perturbations, characteristic points, convection-dominated problems, parabolic boundary layers.

UDC: 517.95

MSC: 35B25, 35C20, 76D05, 76D10

Received: 25.10.2013

DOI: 10.4213/rm9584


 English version:
Russian Mathematical Surveys, 2014, 69:3, 435–480

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© Steklov Math. Inst. of RAS, 2024