Abstract

We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem $\partial _tu+Xu=f, u_{\vert _ {t=0}}=g,$ where $X$ is the vector fieldwith a boundedness condition on the divergence of each vector field $a_1,a_2$. This model was studied in the paper [LL] with a $W^{1,1}$ regularity assumption replacing our $BV$ hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of [Am] and we combine some consequences of the Alberti rank-one structure theorem for $BV$ vector fields with a regularization procedure. Our regularization kernel is not restricted to be a convolution and is introduced as an unknown function. Our method amounts to commute a pseudo-differential operator with a $BV$ function