Failure of the Chain Rule in the Non Steady Two-Dimensional Setting

In Crippa et al. (Ann. Sc. Norm. Super. Pisa Cl. Sci. XVII:1–18, 2017), 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 b : ℝ d → ℝ d $$\mathbf b \colon \mathbb {R}^d \to \mathbb {R}^d$$ , d ≥ 3. By the analysis of Bianchini and Gusev (Arch. Ration. Mech. Anal. 222:451–505, 2016) the assumption d ≥ 3 is essential, as in the two dimensional setting, under the further assumption b≠0 a.e., the Hamiltonian structure prevents from constructing renormalization defects. In this note, following the ideas of Bianchini et al. (SIAM J. Math. Anal. 48:1–33, 2016), 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 b : ℝ 2 → ℝ 2 $$\mathbf b \colon \mathbb {R}^2 \to \mathbb {R}^2$$ such that there exists a non trivial, bounded distributional solution u to ∂ t u + div ( u b ) = 0 $$\displaystyle \partial _t u + \operatorname {div}(u\mathbf b) = 0 $$ for which the distribution ∂ t u 2 + div u 2 b $$\partial _t \left (u^2 \right ) + \operatorname {div}\left (u^2 \mathbf {b}\right )$$ is not (representable by) a Radon measure. MSC (2010): 35F05, 35A02, 35Q35