Well-posedness of space and time dependent transport equations on a network

TL;DR

This paper establishes the well-posedness of linear transport equations with space and time dependent velocities on networks, proving existence, uniqueness, and stability of weak solutions using renormalization techniques.

Contribution

It extends the theory of transport equations to non-smooth, space-time dependent velocities on networks, including existence, uniqueness, and continuous dependence results.

Findings

Proved existence and uniqueness of weak solutions.

Demonstrated continuous dependence on data and velocity.

Applied results to the continuity equation on networks.

Abstract

This article is concerned with the study of weak solutions of a linear transport equation on a bounded domain with coupled boundary data for general non smooth space and time dependent velocity fields. The existence of solutions, its uniqueness and the continuous dependence of the solution on the initial and boundary data as well on the velocity is proven. The results are based on the renormalization property. At the end, the theory is shown to be applicable to the continuity equation on a network.