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

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\colon (0,T) \times \mathbb R^d \to \mathbb R^d$, $T>0$. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system). It is well known that in the generic multi-dimensional case ($d\ge 1$) 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 $b$ (e.g. Sobolev regularity) are needed in order to obtain uniqueness. We prove that in the one-dimensional case ($d=1$) near incompressibility is sufficient for existence and uniqueness of locally integrable weak solutions. We also study compactness properties of the associated Lagrangian flows.