On existence and uniqueness of weak solutions to nonlocal conservation laws with BV kernels

In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution γ ∗ q , we weaken the standard assumption on the kernel γ ∈ L ∞ ( ( 0 , T ) ; W 1 , ∞ ( R ) ) to the substantially more general condition γ ∈ L ∞ ( ( 0 , T ) ; B V ( R ) ) , which allows for discontinuities in the kernel.