*Published Paper*

**Inserted:** 12 oct 2017

**Last Updated:** 23 dec 2019

**Journal:** Current Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg

**Pages:** 33-60

**Year:** 2018

**Doi:** 10.1007/978-3-319-89800-1_2

**Abstract:**

In CGSW1, the authors provide, via an abstract convex integration method, a vast class of counterexamples to the chain rule problem for the divergence operator applied to bounded, autonomous vector fields in $\mathbf b \colon \mathbb R^d \to \mathbb R^d$, $d\ge 3$.
By the analysis of BG the assumption $d \ge 3$ is essential, as in the two dimensional setting, under the further assumption $\mathbf b \ne 0$ a.e., the Hamiltonian structure prevents from constructing renormalization defects. In this note, following the ideas of BBG, we complete the analysis, by considering the non-steady, two dimensional case: we show that it is possible to construct a bounded, autonomous, divergence-free vector field $\mathbf b \colon \mathbb R^2 \to \mathbb R^2$ such that there exists a non trivial, bounded distributional solution $u$ to
\[
\partial_t u + \text{div}(u\mathbf b) = 0
\]
for which the distribution $\partial_t \left(u^2 \right) + \text{div}\left(u^2 \mathbf{b}\right)$ is *not* (representable by) a Radon measure.

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