A current based approach for the uniqueness of the continuity equation

Abstract: We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field $u \in [L^1 (0,T; W^{1,p}(\mathbb{T}^d)) \cap L^\infty ((0,T) \times \mathbb{T}^d)]^d$ with $\operatorname{div}(u) ^- \in L^1 (0,T; L^\infty (\mathbb{T}^d))$ and an initial datum $\rho_0 \in L^q (\mathbb{T}^d)$, where $\mathbb{T}^d$ is the $d$-dimensional torus and $ 1 \leq p,q \leq +\infty$ such that $1/p + 1/q =1$ 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.