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 ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε 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 O ( ε ). Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O ( ε − 1 ) is considered. The novelty of this article is the case of α < 1, which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 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.