DCDS
Boundary layers in smooth curvilinear domains: Parabolic problems
Gung-Min Gie Makram Hamouda Roger Témam
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 ε is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to ε, 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 ε and ε1/2 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.
keywords: heat equation singular perturbations boundary layers curvilinear coordinates.
NHM
Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary
Gung-Min Gie Makram Hamouda Roger Temam
We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order $ε^j$, $j=0,1$, where $ε$ is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $ε^j$, $j=0,1$, for $ε$ small . Using the asymptotic expansion, we prove that the Navier-Stokes solutions converge, as the viscosity parameter tends to zero, to the corresponding Euler solution in the natural energy norm. This work generalizes earlier results in [14] or [26], which discussed the case of a channel domain, while here the domain is curved.
keywords: curvilinear coordinates. Navier-Stokes equations Boundary layers singular perturbations
DCDS
Recent progresses in boundary layer theory
Gung-Min Gie Chang-Yeol Jung Roger Temam
In this article, we review recent progresses in boundary layer analysis of some singular perturbation problems. Using the techniques of differential geometry, an asymptotic expansion of reaction-diffusion or heat equations in a domain with curved boundary is constructed and validated in some suitable functional spaces. In addition, we investigate the effect of curvature as well as that of an ill-prepared initial data. Concerning convection-diffusion equations, the asymptotic behavior of their solutions is difficult and delicate to analyze because it largely depends on the characteristics of the corresponding limit problems, which are first order hyperbolic differential equations. Thus, the boundary layer analysis is performed on relatively simpler domains, typically intervals, rectangles, or circles. We consider also the interior transition layers at the turning point characteristics in an interval domain and classical (ordinary), characteristic (parabolic) and corner (elliptic) boundary layers in a rectangular domain using the technique of correctors and the tools of functional analysis. The validity of our asymptotic expansions is also established in suitable spaces.
keywords: corner layers. initial layers curvilinear coordinates turning points Boundary layers singular perturbations

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