Boundary layers in smooth curvilinear domains: Parabolic problems

The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in $\mathbb{R}^{d}$, $d \geq 2$, when the diffusivity parameter e is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to e, of the heat solution is obtained at all orders. It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of e and e1/2 must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.