Steady Nearly Incompressible Vector Fields in Two-Dimension: 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 ${{\rm div}(\beta(u)b)}$ in terms of div b and div(ub). 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. Malý, up to an error term which is a measure concentrated on the so-called 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 the case when d = 2, we provide complete characterization of div(${\beta(u)b}$) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on ${{\mathbb{R}^{2}}}$ 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 renormalized, that is when it also solves ${\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}$ for any smooth function ${\beta \colon{\mathbb{R}} \to {\mathbb{R}}}$. As a consequence, we obtain new a uniqueness result for this equation.