International Journal of Differential Equations
Volume 2013 (2013), Article ID 532987, 13 pages
Research Article

Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

1Department of Mathematics, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA
2Ulsan National Institute of Science and Technology, San 194, Banyeon-ri, Eonyang-eup, Ulju Gun, Ulsan 689-798, Republic of Korea
3The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, IN 47405, USA

Received 27 January 2013; Accepted 1 April 2013

Academic Editor: Norio Yoshida

Copyright © 2013 Gung-Min Gie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.