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 : (0; T)×Rd → Rd, T > 0. 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 (d ≥ 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 ows. © 2019 American Institute of Mathematical Sciences. All Rights Reserved.