Boundary layer theory for convection-diffusion equations in a circle

Abstract

This paper is devoted to boundary layer theory for singularly perturbed convection-diffusion equations in the unit circle. Two characteristic points appear, (±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.