SFB 1313 Publication "Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks"

New SFB 1313 publication, published in the "Journal of Mathematical Analysis and Applications". The work has been developed within the SFB 1313 research project  B03 and SFB 1313 internal research project I-08.

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

Authors

• Taras Mel'nyk (Taras University Ukranie, SFB 1313 associated reseacher, internal research project I-08)
• Christian Rohde (University of Stuttgart, SFB 1313 research projects B03 and C02)

Abstract

We consider time-dependent convection-diffusion problems with high Péclet number of order O(ε−1) in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order O(ε). 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 O(εα), α>0.
The asymptotic behaviour of the solution is studied as ε→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 α: α=1, α>1, and α∈(0,1). We construct the asymptotic approximation of the solution, which provides us with the hyperbolic limit model for ε→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.