Initial-boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

TL;DR

This paper proves existence, uniqueness, and stability of solutions for one-dimensional transport equations with bounded, nearly incompressible velocity fields, introducing new methods for the general case and analyzing regularity and counterexamples.

Contribution

It introduces a novel construction for handling general velocity fields in transport equations, extending previous techniques to irregular settings.

Findings

Existence and uniqueness of solutions under boundedness assumptions.

Stability of solutions in weak and strong topologies.

Counterexample for sign-changing velocity fields.

Abstract

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a "lagrangian formulation" of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the $BV$ regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a $BV$-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property.