Abstract
In this chapter, we present some recent progresses, which are based on [GJT16], about the boundary layer analysis in a domain enclosed by a curved boundary.
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Notes
- 1.
In space dimension 3 (and 2), the Laplacian (Laplace-Beltrami operator) of a vector field \(\mathbf{v}\) is defined by the identity \(\varDelta \mathbf{v} = \nabla (\text{div}\,\mathbf{v}) -\text{curl}(\text{curl}\,\mathbf{v})\); see, e.g., [Cia05, Kli78, Bat99]. We know that other definitions of the Laplacian of a vector, which possess different properties, are used in different contexts; see, e.g., [Cia05, Kli78].
- 2.
Here \({\boldsymbol \delta }_{\varGamma }\) is used to denote the delta measure on Γ and it should not be confused with the (“small”) number δ used at other places in the text.
- 3.
Here again \({\boldsymbol \delta }_{\varGamma }\) denotes the delta measure supported on the boundary Γ and is not related to the “small” coefficient δ.
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Gie, GM., Hamouda, M., Jung, CY., Temam, R.M. (2018). Boundary Layers in a Curved Domain in \(\mathbb{R}^{d}\), d = 2, 3. In: Singular Perturbations and Boundary Layers. Applied Mathematical Sciences, vol 200. Springer, Cham. https://doi.org/10.1007/978-3-030-00638-9_3
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