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.
Categories: math.AP