Interaction of boundary layers and corner singularities
Chang-Yeol Jung Roger Temam
Our aim in this article is to study the interaction of \textit{boundary layers} and \textit{corner singularities} in the context of singularly perturbed convection-diffusion equations. For the problems under consideration, we determine a simplified form of the corner singularities and show how to use it for the numerical approximation of such problems in the context of variational approximations using the concept of \textit{enriched spaces}.
keywords: boundary layers; boundary layer element; finite elements; singularly perturbed problem; convection-diffusion
Wave propagation in random waveguides
Chang-Yeol Jung Alex Mahalov
We study uncertainty bounds and statistics of wave solutions through a random waveguide which possesses certain random inhomogeneities. The waveguide is composed of several homogeneous media with random interfaces. The main focus is on two homogeneous media which are layered randomly and periodically in space. Solutions of stochastic and deterministic problems are compared. The waveguide media parameters pertaining to the latter are the averaged values of the random parameters of the former. We investigate the eigenmodes coupling due to random inhomogeneities in media, i.e. random changes of the media parameters. We present an efficient numerical method via Legendre Polynomial Chaos expansion for obtaining output statistics including mean, variance and probability distribution of the wave solutions. Based on the statistical studies, we present uncertainty bounds and quantify the robustness of the solutions with respect to random changes of interfaces.
keywords: Evolution of probability distribution; Monte Carlo simulation; Stochastic partial differential equation; random media; random interface.
Boundary layers for the 2D linearized primitive equations
Makram Hamouda Chang-Yeol Jung Roger Temam
In this article, we establish the asymptotic behavior, when the viscosity goes to zero, of the solutions of the Linearized Primitive Equations (LPEs) in space dimension $2$. More precisely, we prove that the LPEs solution behaves like the corresponding inviscid problem solution inside the domain plus an explicit corrector function in the neighborhood of some parts of the boundary. Two cases are considered, the subcritical and supercritical modes depending on the fact that the frequency mode is less or greater than the ratio between the reference stratified flow (around which we linearized) and the buoyancy frequency. The problem of boundary layers for the LPEs is of a new type since the corresponding limit problem displays a set of (unusual) nonlocal boundary conditions.
keywords: Primitive equations boundary layers singular perturbation analysis
Asymptotic analysis for the 3D primitive equations in a channel
Makram Hamouda Chang-Yeol Jung Roger Temam
In this article, we give an asymptotic expansion, with respect to the viscosity which is considered here to be small, of the solutions of the $3D$ linearized Primitive Equations (EPs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice of the non-local boundary conditions for the non-viscous PEs.
keywords: Primitive equations boundary layers singular perturbation analysis.
Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers
Daniel Bouche Youngjoon Hong Chang-Yeol Jung

We study the asymptotic behavior of the two dimensional Helmholtz scattering problem with high wave numbers in an exterior domain, the exterior of a circle. We impose the Dirichlet boundary condition on the obstacle, which corresponds to an incidental wave. For the outer boundary, we consider the Sommerfeld conditions. Using a polar coordinates expansion, the problem is reduced to a sequence of Bessel equations. Investigating the Bessel equations mode by mode, we find that the solution of the scattering problem converges to its limit solution at a specific rate depending on k.

keywords: Electromagnetism acoustics Helmholtz equations scattering problem asymptotic analysis
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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