Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

Box 1

The different correctors.

𝜑^𝑗 =𝜑^𝑗_𝑙+𝜑^𝑗_𝑢 is the parabolic boundary layer corrector near 𝑦=0 and 𝑦=1,

𝜉^𝑗=𝜉^𝑗_𝑙+𝜉^𝑗_𝑢 is the elliptic corrector which resolves the compatibility issues, in the construction of 𝜑^𝑗, at the corners (1,0) and (1,1),

𝜃^𝑗 is the ordinary boundary layer corrector near 𝑥 = 0,

𝜁^𝑗=𝜁^𝑗_𝑙+𝜁^𝑗_𝑢  is the ordinary corner layer corrector which manages the effect of 𝜑^𝑗 along 𝑥=0,

𝜂^𝑗=𝜂^𝑗_𝑙+𝜂^𝑗_𝑢  is the supplementary corrector that manages the effects of 𝜃^𝑗 and 𝜁^𝑗 along the edges 𝑦=0 and 𝑦=1.