A current based approach for the uniqueness of the continuity equation

We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field $u\in [L^1(0,T;W^{1,p}({\mathbb{T}}^d))\cap L^{\mathrm{\infty }}((0,T)\times {\mathbb{T}}^d){]}^d$ with $\mathrm{div}(u{)}^{-}\in L^1(0,T;L^{\mathrm{\infty }}({\mathbb{T}}^d))$ and an initial datum ${\rho }_0\in L^q({\mathbb{T}}^d)$, where ${\mathbb{T}}^d$ is the $d$-dimensional torus and $1\le p,q\le +\mathrm{\infty }$ such that $1\mathrm{/}p+1\mathrm{/}q=1$ without using the theory of renormalized solutions. We propose a more geometric approach which will however still rely on a strong $L^1$ estimate on the commutator (which is the key technical tool when using renormalized solutions, too), but other than that will be based on the theory of currents.