A Note on the Construction of Function Spaces for Distributed-Microstructure Models with Spatially Varying Cell Geometry
Year: 2008
International Journal of Numerical Analysis and Modeling, Vol. 5 (2008), Iss. 5 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2008-IJNAM-843
International Journal of Numerical Analysis and Modeling, Vol. 5 (2008), Iss. 5 : pp. 109–125
Published online: 2008-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 17
Keywords: Lebesgue spaces Sobolev spaces distributed-microstructure model direct integral reaction–diffusion homogenisation.