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2018 | OriginalPaper | Buchkapitel

3. Boundary Layers in a Curved Domain in \(\mathbb{R}^{d}\), d = 2, 3

verfasst von : Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam

Erschienen in: Singular Perturbations and Boundary Layers

Verlag: Springer International Publishing

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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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Vorheriges Kapitel Singular Perturbations in Higher Dimensions in a Channel

Nächstes Kapitel Corner Layers and Turning Points for Convection-Diffusion Equations

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].

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.

Here again \({\boldsymbol \delta }_{\varGamma }\) denotes the delta measure supported on the boundary Γ and is not related to the “small” coefficient δ.

[Bat99]

G. K. Batchelor. An introduction to fluid dynamics. Cambridge Mathematical Library. Cambridge University Press, Cambridge, paperback edition, 1999.MATH

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John Rozier Cannon. The one-dimensional heat equation, volume 23 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 1984. With a foreword by Felix E. Browder.

[Cia05]

Philippe G. Ciarlet. An introduction to differential geometry with application to elasticity. J. Elasticity, 78/79(1–3):iv+215, 2005. With a foreword by Roger Fosdick.

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[GHT12]

Gung-Min Gie, Makram Hamouda, and Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Netw. Heterog. Media, 7(4):741–766, 2012.MathSciNetCrossRef

[GHT10b]

Gung-Min Gie, Makram Hamouda, and Roger Temam. Boundary layers in smooth curvilinear domains: parabolic problems. Discrete Contin. Dyn. Syst., 26(4):1213–1240, 2010.MathSciNetMATH

[GJT16]

Gung-Min Gie, Chang-Yeol Jung, and Roger Temam. Recent progresses in boundary layer theory. Discrete Contin. Dyn. Syst., 36(5):2521–2583, 2016.MathSciNetMATH

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Gung-Min Gie and James P. Kelliher. Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions. J. Differential Equations, 253(6):1862–1892, 2012.MathSciNetCrossRef

[GKLMN18]

Gung-Min Gie, James P. Kelliher, M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes. Vanishing viscosity limit of some symmetric flows. Preprint, 2018.

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Chang-Yeol Jung, Eunhee Park, and Roger Temam. Boundary layer analysis of nonlinear reaction-diffusion equations in a smooth domain. Adv. Nonlinear Anal., 6(3):277–300, 2017MathSciNetMATH

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Chang-Yeol Jung, Eunhee Park, and Roger Temam. Boundary layer analysis of nonlinear reaction-diffusion equations in a polygonal domain. Nonlinear Anal., 148:161–202, 2017.MathSciNetCrossRef

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James P. Kelliher. Vanishing viscosity and the accumulation of vorticity on the boundary. Commun. Math. Sci., 6(4):869–880, 2008.MathSciNetCrossRef

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M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, and Michael Taylor. Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows. Bull. Braz. Math. Soc. (N.S.), 39(4):471–513, 2008.MathSciNetCrossRef

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Jeffrey B. Rauch and Frank J. Massey III. Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Amer. Math. Soc., 189 (1974), 303–318.MathSciNetMATH

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Stephen Smale. Smooth solutions of the heat and wave equations. Comment. Math. Helv., 55(1):1–12, 1980.MathSciNetCrossRef

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Titel: Boundary Layers in a Curved Domain in , d = 2, 3
verfasst von: Gung-Min Gie
Makram Hamouda
Chang-Yeol Jung
Roger M. Temam
Verlag: Springer International Publishing
Buch: Singular Perturbations and Boundary Layers
Print ISBN: 978-3-030-00637-2

Electronic ISBN: 978-3-030-00638-9

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-030-00638-9_3

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