Renormalized solutions to the continuity equation with an integrable damping term

We consider the continuity equation with a nonsmooth vector field and a damping term. In their fundamental paper, DiPerna and Lions (Invent Math 98:511–547, 1989) proved that, when the damping term is bounded in space and time, the equation is well posed in the class of distributional solutions and the solution is transported by suitable characteristics of the vector field. In this paper, we prove existence and uniqueness of renormalized solutions in the case of an integrable damping term, employing a new logarithmic estimate inspired by analogous ideas of Ambrosio et al. (Rendiconti del Seminario Fisico Matematico di Padova 114:29–50, 2005), Crippa and De Lellis (J Reine Angew Math 616:15–46, 2008) in the Lagrangian case.