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

TL;DR

This paper characterizes the divergence of composite functions involving nearly incompressible vector fields in 2D, providing new insights into their structure, renormalization, and uniqueness of solutions for related PDEs.

Contribution

It offers a complete characterization of divergence for such vector fields in 2D and introduces new conditions ensuring renormalization and uniqueness of solutions.

Findings

Constructed examples with nonzero tangential measure term.

Provided complete divergence characterization in 2D for nearly incompressible BV fields.

Established new sufficient conditions for solution renormalization and uniqueness.

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.