Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks

TL;DR

This paper analyzes the asymptotic behavior of high Péclet number convection-diffusion problems in thin graph-like networks, deriving hyperbolic limit models and uniform estimates as the network shrinks.

Contribution

It introduces a detailed asymptotic analysis for nonlinear convection-diffusion in thin networks, identifying different regimes based on boundary inhomogeneity strength.

Findings

Asymptotic approximation constructed for all cases.

Hyperbolic limit models derived for different parameter regimes.

Uniform pointwise and energy estimates established.

Abstract

We consider time-dependent convection-diffusion problems with high P\'eclet number of order $\mathcal{O}(\varepsilon^{-1})$ in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order $\mathcal{O}(\varepsilon).$ On the lateral surfaces of the thin cylinders and the boundaries of the nodes we account for solution-dependent inhomogeneous Robin boundary conditions which can render the associated initial-boundary problem to be nonlinear. The strength of the inhomogeneity is controlled by an intensity factor of order ${\mathcal{O}} (\varepsilon^\alpha)$, $\alpha >0$. The asymptotic behaviour of the solution is studied as $\varepsilon \to 0,$ i.e., when the diffusion coefficients are eliminated and the thin three-diemnsional network is shrunk into a graph. There are three qualitatively different cases in the asymptotic behaviour of the solution depending on the value of the intensity parameter $\alpha:$ $\alpha =1,$ $\alpha > 1,$ and $\alpha \in (0, 1).$ We construct the asymptotic approximation of the solution, which provides us with the hyperbolic limit model for $\varepsilon \to 0$ for the first two cases, and prove the corresponding uniform pointwise estimates and energy estimates. As the main result, we derive uniform pointwise estimates for the difference between the solutions of the convection-diffusion problem and the zero-order approximation that includes the solution of the corresponding hyperbolic limit problem.