Steady nearly incompressible vector fields in 2D: chain rule and renormalization

Given bounded vector field $bcolon RR^d o RR^d$, scalar field $ucolon RR^d o RR$ and a smooth function $eta colon RR o RR$ we study the characterization of the distribution $div(eta(u)b)$ in terms of $div b$ and $div(u b)$. In the case of $BV$ vector fields $b$ (and under some further assumptions) such characterization was obtained by L. Ambrosio, C. De Lellis and J. Mal'y, up to an error term which is a measure concentrated on so-called emph{tangential set} of $b$. We answer some questions posed in their paper concerning the properties of this term. In particular we construct a nearly incompressible $BV$ vector field $b$ and a bounded function $u$ for which this term is nonzero. For steady nearly incompressible vector fields $b$ (and under some further assumptions) in case when $d=2$ we provide complete characterization of $div(eta(u) b)$ in terms of $div b$ and $div(u b)$. Our approach relies on the structure of level sets of Lipschitz functions on $RR^2$ obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique we obtain new sufficient conditions when any bounded weak solution $u$ of $d_t u + b cdot abla u=0$ is emph{renormalized}, i.e. also solves $d_t eta(u) + b cdot abla eta(u)=0$ for any smooth function $eta colonRR o RR$. As a consequence we obtain new uniqueness result for this equation.