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In this paper, we study linear parabolic equations on a finite oriented star-shaped network; the equations are coupled by transmission conditions set at the inner node, which do not impose continuity on the unknown. We consider this problem as a parabolic approximation of a set of the first-order linear transport equations on the network, and we prove that when the diffusion coefficient vanishes, the family of solutions converges to the unique solution to the first-order equations satisfying suitable transmission conditions at the inner node, which are determined by the parameters appearing in the parabolic transmission conditions.

Vanishing viscosity approximation for linear transport equations on finite star-shaped networks