Renormalized solutions to the continuity equation with an integrable damping term

2014/11/03 by Maria Colombo, Colombo, Maria, Gianluca Crippa +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1411.0451

openalex publication_date 2014/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• On functions of bounded mean oscillation

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