We consider a vector field whose coefficients are functions of bounded variation, with a bounded divergence. We prove the uniqueness of continuous solutions for the Cauchy problem.
"Uniqueness of continuous solutions for BV vector fields." Duke Math. J. 111 (2) 357 - 384, 1 February 2002. https://doi.org/10.1215/S0012-7094-01-11126-5