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

Given bounded vector field b : R → R, scalar field u : R → R and a smooth function β : R → R we study the characterisation of the distribution div(β(u)b) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions) such characterisation was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on 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 case when d = 2 we provide complete characterisation of div(β(u)b) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on R obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique we obtain new sufficient conditions when any bounded weak solution u of ∂tu+ b ·∇u = 0 is renormalized, i.e. also solves ∂tβ(u) + b ·∇β(u) = 0 for any smooth function β : R→ R. As a consequence we obtain new uniqueness result for this equation.