Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications

This paper provides an overview of the theory of flows associated to nonsmooth vector fields, describing the main developments from the seminal paper (DiPerna and Lions in Invent Math 98:511–547, 1989) till now. The problem of well posedness of ODE’s associated to vector fields arises in many fields, for instance conservation laws (via the theory of characteristics) and fluid mechanics (when looking for consistence between Eulerian and Lagrangian points of view). The theory developed so far covers many classes of vector fields and, besides uniqueness, also more quantitative aspects, as stability estimates and differentiability of the flow. Detailed lecture notes on this topic are given in Ambrosio (in: Dacorogna, Marcellini (eds) Lecture Notes in Mathematics “Calculus of variations and non-linear partial differential equations” (CIME Series, Cetraro, 2005), vol 1927, pp 2–41, 2008), Ambrosio and Crippa (Lect Notes UMI 5:3–54, 2008), Ambrosio and Crippa (Proc R Soc Edinb Sect A 144:1191–1244 2014), Ambrosio and Trevisan (Ann Fac Sci Toulouse, 2016).