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

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.Comment: 50 pages, 8 figure

A connection between viscous profiles and singular ODEs

We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems. We focus, in particular, on the case of the compressible Navier Stokes equation in one space variable written in Eulerian coordinates. We describe the link between these profiles and a singular ordinary differential equation in the form $$dV / dt = F(V) / z (V) .$$ Here $V \in R^d$ and the function F takes values into $R^d$ and is smooth. The real valued function z is as well regular: the equation is singular in the sense that z (V) can attain the value 0.Comment: 6 pages, minor change