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 $\alpha$ in the boundary condition $\varepsilon \partial_{\boldsymbol{\nu}_\varepsilon} u_\varepsilon - u_\varepsilon \, \overrightarrow{V_\varepsilon}\boldsymbol{\cdot}\boldsymbol{\nu}_\varepsilon = \varepsilon^{\alpha} \varphi_\varepsilon $ 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 $\mathcal{O}(\varepsilon).$ Inside of the thin network a time-dependent convection-diffusion equation with high P\'eclet number of order $\mathcal{O}(\varepsilon^{-1})$ is considered. The novelty of this article is the case of $\alpha <1,$ which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity $\varphi_\varepsilon$ (the cases $\alpha =1$ and $\alpha >1$ were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution $u_\varepsilon$ as $\varepsilon \to 0,$ 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 $\varepsilon.$