Failure of the chain rule for the divergence of bounded vector fields

TL;DR

This paper constructs numerous counterexamples demonstrating the failure of the chain rule for divergence in bounded vector fields in three dimensions, revealing complex renormalization defects.

Contribution

It introduces a convex integration method to generate diverse divergence defects, highlighting the breakdown of the chain rule in this context.

Findings

Counterexamples with absolutely continuous defects

Counterexamples with non-measure defects

Demonstration of chain rule failure in bounded vector fields

Abstract

We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to Lebesgue measure, or defects which are not even measures.