Vanishing viscosity approximation for linear transport equations on finite starshaped networks

TL;DR

This paper investigates the vanishing viscosity limit for linear transport equations on finite star-shaped networks, proving convergence of solutions of parabolic approximations to the first order problem with specific transmission conditions.

Contribution

It establishes the convergence of solutions of parabolic equations to first order transport equations on networks, characterizing the limiting transmission conditions.

Findings

Solutions of parabolic equations converge to first order solutions as viscosity vanishes

Transmission conditions at the network node are derived from the parabolic approximation

The limiting solution is unique and satisfies specific transmission conditions

Abstract

In this paper we study linear parabolic equations on a finite oriented starshaped 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 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 and satisfies suitable transmission conditions at the inner node, which are determined by the parameters appearing in the parabolic transmission conditions.