  We face the well-posedness of linear transport Cauchy problems
$$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u =
f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb
R}^n\end{cases}$$ under borderline integrability assumptions on the divergence
of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying
$\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and
$$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T;
\operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence
and uniqueness of weak solutions. Moreover, optimality is shown in the
following way: for every $\gamma>1$, we construct an example of a bounded
autonomous velocity field $b$ with $$\operatorname{div} b\in
\operatorname{Exp}\left(\frac{L}{\log^\gamma L}\right) ,$$ for which the
associate Cauchy problem for the transport equation admits infinitely many
solutions. Stability questions and further extensions to the $BV$ setting are
also addressed.
