  We construct infinitely many incompressible Sobolev vector fields $u \in C_t
W^{1,\tilde p}_x$ on the periodic domain $\mathbb{T}^d$ for which uniqueness of
solutions to the transport equation fails in the class of densities $\rho \in
C_t L^p_x$, provided $1/p + 1/\tilde p > 1 + 1/d$. The same result applies to
the transport-diffusion equation, if, in addition $p'<d$.
