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

Abstract

Given bounded vector field $b : \mathbb R^d \to \mathbb R^d$, scalar field $u : \mathbb R^d \to \mathbb R$ and a smooth function $\beta : \mathbb R \to \mathbb R$ we study the characterization of the distribution $\mathrm{div}(\beta(u)b)$ in terms of $\mathrm{div}\, b$ and $\mathrm{div}(u b)$. 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 $\mathrm{div}(\beta(u) b)$ in terms of $\mathrm{div}\, b$ and $\mathrm{div}(u b)$. Our approach relies on the structure of level sets of Lipschitz functions on $\mathrm R^2$ obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique we obtain new sufficient conditions when any bounded weak solution $u$ of $\partial_t u + b \cdot \nabla u=0$ is \emph{renormalized}, i.e. also solves $\partial_t \beta(u) + b \cdot \nabla \beta(u)=0$ for any smooth function $\beta : \mathbb R \to \mathbb R$. As a consequence we obtain new uniqueness result for this equation.