Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners. (9th May 2013) - Gie, Gung-Min; Jung, Chang-Yeol; Temam, Roger
Abstract : We study the asymptotic behavior at small diffusivity of the solutions, u ε, to a convection-diffusion equation in a rectangular domainΩ . 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∂ Ω, betweenu ε and the corresponding limit solution, u 0, we propose asymptotic expansions ofu ε at any arbitrary, but fixed, order. In order to manage some singular effects near the four corners ofΩ, 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 ofu ε and the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces.