  We establish local-in-time existence and uniqueness results for nonlocal
conservation laws in several space dimensions under weak (that is, Sobolev or
BV) differentiability assumptions on the convolution kernel. In contrast to the
case of a smooth kernel, in general the solution experiences finite-time
blow-up. We provide an explicit example showing that solutions corresponding to
different smooth approximations of the convolution kernel in general converge
to different measures after the blow-up time. This rules out a fairly natural
strategy for extending the notion of solution of the nonlocal conservation law
after the blow-up time.
