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Abstract
In this article,we review recent progresses in boundary layer analysis of somesingular perturbation problems.Using the techniques of differential geometry,an asymptotic expansion of reaction-diffusion or heat equations in a domain with curved boundaryis constructed and validated in some suitable functional spaces.In addition, we investigate the effect of curvatureas well as thatof an ill-prepared initial data.Concerning convection-diffusion equations, the asymptotic behavior of their solutionsis 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 domainusing the technique of correctors and the tools of functional analysis.The validity of our asymptotic expansions is also established in suitable spaces.
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