An Overview on Some Results Concerning the Transport Equation and its Applications to Conservation Laws

TL;DR

This paper provides an informal overview of the theory of transport equations with non-smooth velocity fields and explores their applications to the well-posedness of hyperbolic conservation laws.

Contribution

It offers a comprehensive summary of recent results on transport equations with irregular velocities and their implications for conservation law problems.

Findings

Insights into the well-posedness of hyperbolic systems

Connections between transport equations and conservation laws

Summary of recent theoretical developments

Abstract

We provide an informal overview on the theory of transport equations with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.