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 - <p style="text-align: justify;">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.</p>