Non-renormalized solutions to the continuity equation

TL;DR

This paper demonstrates that certain continuous, incompressible vector fields with limited regularity can lead to non-unique solutions of the continuity equation, challenging assumptions about solution uniqueness.

Contribution

It provides explicit examples of non-uniqueness in the continuity equation for vector fields with low regularity, expanding understanding of solution behavior.

Findings

Existence of continuous, $W^{1,p}$ vector fields with $p<d-1$ causing non-uniqueness.

Non-uniqueness occurs despite the vector fields being incompressible.

Results challenge previous beliefs about solution uniqueness under regularity constraints.

Abstract

We show that there are continuous, $W^{1,p}$ ($p<d-1$), incompressible vector fields for which uniqueness of solutions to the continuity equation fails.