Well-posedness for the continuity equation for vector fields with suitable modulus of continuity

We prove well-posedness of linear scalar conservation laws using only assumptions on the growth and the modulus of continuity of the velocity field, but not on its divergence. As an application, we obtain uniqueness of solutions in the atomic Hardy space, $H^1$, for the scalar conservation law induced by a class of vector fields whose divergence is an unbounded BMO function.