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Volume 5, Issue 5
A Note on the Construction of Function Spaces for Distributed-Microstructure Models with Spatially Varying Cell Geometry

S. Meier & M. Böhm

Int. J. Numer. Anal. Mod., 5 (2008), pp. 109-125.

Published online: 2018-11

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  • Abstract

We construct Lebesgue and Sobolev spaces of functions defined on a continuous distribution of domains {$Y_x \subset \mathbb{R}^m$ : $x \in \Omega$}. The resulting spaces can be viewed as a generalisation of the Bochner spaces $L_p(\Omega;W_q^l(Y))$ for the case that $Y$ depends on $x \in \Omega$. Furthermore, we introduce a Lebesgue space of functions defined on the boundaries {$∂Y_x : x \in \Omega$}. The latter construction relies on a uniform Lipschitz parametrisation of the above collection of boundaries, interpreted as a higher-dimensional manifold. The results are applied to prove existence, uniqueness and upper and lower bounds for a distributed-microstructure model of reactive transport in a heterogeneous porous medium.

  • Keywords

Lebesgue spaces, Sobolev spaces, distributed-microstructure model, direct integral, reaction–diffusion, homogenisation.

  • AMS Subject Headings

46E30, 46E35, 35K57, 35B27

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-5-109, author = {Meier , S. and Böhm , M.}, title = {A Note on the Construction of Function Spaces for Distributed-Microstructure Models with Spatially Varying Cell Geometry}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {5}, number = {5}, pages = {109--125}, abstract = {

We construct Lebesgue and Sobolev spaces of functions defined on a continuous distribution of domains {$Y_x \subset \mathbb{R}^m$ : $x \in \Omega$}. The resulting spaces can be viewed as a generalisation of the Bochner spaces $L_p(\Omega;W_q^l(Y))$ for the case that $Y$ depends on $x \in \Omega$. Furthermore, we introduce a Lebesgue space of functions defined on the boundaries {$∂Y_x : x \in \Omega$}. The latter construction relies on a uniform Lipschitz parametrisation of the above collection of boundaries, interpreted as a higher-dimensional manifold. The results are applied to prove existence, uniqueness and upper and lower bounds for a distributed-microstructure model of reactive transport in a heterogeneous porous medium.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/843.html} }
TY - JOUR T1 - A Note on the Construction of Function Spaces for Distributed-Microstructure Models with Spatially Varying Cell Geometry AU - Meier , S. AU - Böhm , M. JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 109 EP - 125 PY - 2018 DA - 2018/11 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/843.html KW - Lebesgue spaces, Sobolev spaces, distributed-microstructure model, direct integral, reaction–diffusion, homogenisation. AB -

We construct Lebesgue and Sobolev spaces of functions defined on a continuous distribution of domains {$Y_x \subset \mathbb{R}^m$ : $x \in \Omega$}. The resulting spaces can be viewed as a generalisation of the Bochner spaces $L_p(\Omega;W_q^l(Y))$ for the case that $Y$ depends on $x \in \Omega$. Furthermore, we introduce a Lebesgue space of functions defined on the boundaries {$∂Y_x : x \in \Omega$}. The latter construction relies on a uniform Lipschitz parametrisation of the above collection of boundaries, interpreted as a higher-dimensional manifold. The results are applied to prove existence, uniqueness and upper and lower bounds for a distributed-microstructure model of reactive transport in a heterogeneous porous medium.

S. Meier & M. Böhm. (1970). A Note on the Construction of Function Spaces for Distributed-Microstructure Models with Spatially Varying Cell Geometry. International Journal of Numerical Analysis and Modeling. 5 (5). 109-125. doi:
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