On the failure of the chain rule for the divergence of Sobolev vector fields

Abstract

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d\ge 3$, for which the chain rule problem has a negative answer. In particular, for any renormalization map $\beta$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T=\mathrm{div}h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u\in W^{1,\tilde{p}}$ and a density $\rho \in L^p$ for which $\mathrm{div}(\rho u)=0$ but $\mathrm{div}(\beta (\rho )u)=T$, provided $1/p+1/\tilde{p}\ge 1+1/(d-1)$

24 pages