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: ℝd→ ℝd, 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: ℝ2→ ℝ2 such that there exists a non trivial, bounded distributional solution u to ∂tu+div(ub)=0 for which the distribution ∂t(u2)+div(u2b) is not (representable by) a Radon measure. MSC (2010): 35F05, 35A02, 35Q35.