Renormalized solutions to the continuity equation with an integrable damping term

TL;DR

This paper establishes existence and uniqueness of renormalized solutions to the continuity equation with an integrable damping term, extending classical results to less regular damping functions using a novel logarithmic estimate.

Contribution

It introduces a new logarithmic estimate to prove well-posedness of the continuity equation with integrable damping, broadening the class of admissible damping terms.

Findings

Proves existence of renormalized solutions with integrable damping.

Establishes uniqueness of solutions under these conditions.

Develops a new logarithmic estimate inspired by prior Lagrangian approaches.

Abstract

We consider the continuity equation with a nonsmooth vector field and a damping term. In their fundamental paper, DiPerna and Lions 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, Lecumberry, and Maniglia, Crippa and De Lellis in the Lagrangian case.