On the transport limit of singularly perturbed convection-diffusion problems on networks
Submission Status: | revision under review |
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- We consider singularly perturbed convection-diffusion equations on one-dimensional
networks (metric graphs) as well as the transport problems arising in the
vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that
guarantee conservation of mass as well as dissipation of a mathematical energy which
allows us to prove stability and well-posedness. For single intervals and appropriately
specified initial conditions, it is well-known that the solutions of the convection-diffusion
problem converge to that of the transport problem with order O(sqrt(eps)) in the L1(L2)-
norm with diffusion eps -> 0. In this paper, we prove a corresponding result for problems
on one-dimensional networks. The main difficulty in the analysis is that the number and
type of coupling conditions changes in the singular limit which gives rise to additional
boundary layers at the interior vertices of the network. Since the values of the solution
at these network junctions are not known a-priori, the asymptotic analysis requires a
delicate choice of boundary layer functions that allows to handle these interior layers.