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 L¹ (0, T; W^1, p (Tᵈ) ) L^ ( (0, T) Tᵈ) ᵈ with div (u) ^- L¹ (0, T; L^ (Tᵈ) ) and an initial datum ₀ Lq (Tᵈ), where Tᵈ is the d-dimensional torus and 1 p, q + 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 L¹ 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.