Non-uniqueness for the transport equation with Sobolev vector fields

TL;DR

This paper demonstrates that the linear transport and transport-diffusion equations with divergence-free Sobolev vector fields can have multiple solutions, challenging assumptions of uniqueness in such settings.

Contribution

It constructs a broad class of examples showing non-uniqueness for these equations with Sobolev vector fields, advancing understanding of solution behavior.

Findings

Non-uniqueness examples for transport equations with Sobolev vector fields

Extension to transport-diffusion equations

Implications for well-posedness in Sobolev spaces

Abstract

We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields in Sobolev spaces $W^{1,p}$.