Some new well-posedness results for continuity and transport equations, and applications to the chromatography system

TL;DR

This paper establishes new well-posedness results for continuity and transport equations, including cases with nearly incompressible vector fields and BV norm blow-up, and applies these to chromatography and Keyfitz-Kranzer systems.

Contribution

It provides novel existence and uniqueness theorems for transport equations with challenging vector fields and extends these results to specific conservation law systems.

Findings

Well-posedness results for nearly incompressible vector fields

Existence and uniqueness in strongly continuous solutions

Applications to chromatography and Keyfitz-Kranzer systems

Abstract

We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blow-up of the BV norm at the initial time. We apply these results (valid in any space dimension) to the k x k chromatography system of conservation laws and to the k x k Keyfitz and Kranzer system, both in one space dimension.