Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

We study the asymptotic behavior at small diffusivity of the solutions, , to a convection-diffusion equation in a rectangular domain . The diffusive equation is supplemented with a Dirichlet boundary condition, which is smooth along the edges and continuous at the corners. To resolve the discrepancy, on , between and the corresponding limit solution, , we propose asymptotic expansions of at any arbitrary, but fixed, order. In order to manage some singular effects near the four corners of , the so-called elliptic and ordinary corner correctors are added in the asymptotic expansions as well as the parabolic and classical boundary layer functions. Then, performing the energy estimates on the difference of and the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces. 1. Introduction We consider a singularly perturbed convection-diffusion equation in a rectangular domain : Here is a small but strictly positive diffusivity parameter, and is a given smooth data with , independent of , for some 's as needed in the analysis below. The function is assumed to be continuous on and smooth on each edge of . Namely, defining the restriction of to the edges of as follows: we assume that If these compatibility conditions were not satisfied, some additional considerations would be necessary, which we do not address here. In what follows, we study the asymptotic behavior of the solutions to (1) at small diffusivity . In a very nice related earlier work, [1], the asymptotic behavior of the solutions of a convection-diffusion equation similar to (1) was discussed. More precisely, in a rectangular domain , the authors considered where , are constants and is a given smooth function in . The function , satisfying the analogue version of (3) in , is assumed to be continuous and piecewise smooth on . By constructing, in [1], asymptotic expansions of with respect to a small diffusivity , the boundary layers of (4) were analyzed in a systematic way. The validity of their asymptotic expansions was established using the maximum principle. Using a simple change of variables which maps to , and setting with a suitable , one can transform (4) to (1) where the two diffusivity parameters and , respectively, in (1) and (4) are compatible; that is, is of order one. Hence, via this transformation, our analysis of (1) in this paper is applicable to (4) as well. Our motivations for conducting the present study appear below. In particular we significantly simplify the proofs of [1] and make the study suitable for more general equations or