Uniqueness for weak solutions of the continuity equation on the torus or in the Euclidean space goes back to the theory of Di Perna and Lions for Sobolev vector fields and to Ambrosio in the BV case. In presence of a boundary, the vector field is required to be tangent to it: the usual weak formulation of this condition is not strong enough to ensure uniqueness, while a different notion of “normal trace” suffices. Here we want to compare the two notions of normal trace and explain why one of them is more suitable in our case.