Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Recent progresses in boundary layer theory

Pages: 2521 - 2583, Volume 36, Issue 5, May 2016      doi:10.3934/dcds.2016.36.2521

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Gung-Min Gie - Department of Mathematics, University of Louisville, Louisville, KY 40292, United States (email)
Chang-Yeol Jung - Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 689-798, South Korea (email)
Roger Temam - Department of Mathematics and The Institute, for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, United States (email)

Abstract: 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:  Boundary layers, singular perturbations, curvilinear coordinates, initial layers, turning points, corner layers.
Mathematics Subject Classification:  35B25, 35C20, 76D10, 35K05.

Received: April 2015;      Revised: September 2015;      Available Online: October 2015.