Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: Strong boundary interactions

Abstract

This article completes the study of the influence of the intensity parameter α in the boundary condition $\mathit{\epsilon }{\partial }_{{\mathit{\nu }}_{\mathit{\epsilon }}}{\mathit{u}}_{\mathit{\epsilon }}-{\mathit{u}}_{\mathit{\epsilon }}\overrightarrow{{\mathit{V}}_{\mathit{\epsilon }}}\mathbf{·}{\mathit{\nu }}_{\mathit{\epsilon }}={\mathit{\epsilon }}^{\mathit{\alpha }}{\mathit{\phi }}_{\mathit{\epsilon }}$ given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order $\mathcal{O}\mathrm{(}\mathit{\epsilon }\mathrm{)}$ . Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order $\mathcal{O}\mathrm{(}{\mathit{\epsilon }}^{-1}\mathrm{)}$ is considered. The novelty of this article is the case of $\mathit{\alpha }\mathrm{<}1$ , which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity ${\mathit{\phi }}_{\mathit{\epsilon }}$ (the cases $\mathit{\alpha }=1$ and $\mathit{\alpha }\mathrm{>}1$ were previously studied by the same authors).

A complete Puiseux asymptotic expansion is constructed for the solution ${\mathit{u}}_{\mathit{\epsilon }}$ as $\mathit{\epsilon }\to 0$ , i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε.