Renormalized solutions to the continuity equation with an integrable damping term
Analysis of PDEs
2014-11-04 v1
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.
Cite
@article{arxiv.1411.0451,
title = {Renormalized solutions to the continuity equation with an integrable damping term},
author = {Maria Colombo and Gianluca Crippa and Stefano Spirito},
journal= {arXiv preprint arXiv:1411.0451},
year = {2014}
}