On the one-dimensional continuity equation with a nearly incompressible vector field

Abstract

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field \begin{document} $b\colon (0,T) × \mathbb{R}^d \to \mathbb{R}^d$ \end{document} , \begin{document} $T>0$ \end{document} . This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system, which has applications in nonlinear elasticity theory). It is well known that in the generic multi-dimensional case ( \begin{document}$d≥ 1$\end{document} ) near incompressibility is sufficient for existence of bounded weak solutions, but uniqueness may fail (even when the vector field is divergence-free), and hence further assumptions on the regularity of \begin{document} $b$ \end{document} (e.g. Sobolev regularity) are needed in order to obtain uniqueness. We prove that in the one-dimensional case ( \begin{document}$d = 1$\end{document} ) near incompressibility is sufficient for existence and uniqueness of locally integrable weak solutions. We also study compactness properties of the associated Lagrangian flows.