Well posedness of ODE's and continuity equations with nonsmooth vector fields, and applications (Q1675688)

The paper gives an overview of the theory of flows associated to nonsmooth vector fields. It describes the main developments since the basic paper by \textit{R. J. DiPerna} and \textit{P. L. Lions} [Invent. Math. 98, No. 3, 511--547 (1989; Zbl 0696.34049)]. Proofs and technical details can be found in a couple of other sources, e.g., in [\textit{B. Dacorogna} (ed.) and \textit{P. Marcellini} (ed.), Calculus of variations and nonlinear partial differential equations. Lectures given at the C. I.M. E. summer school, Cetraro, Italy, June 27--July 2, 2005. With a historical overview by Elvira Mascolo. Berlin: Springer (2008; Zbl 1126.35004)] or in [\textit{L. Ambrosio} and \textit{G. Crippa}, Proc. R. Soc. Edinb., Sect. A, Math. 144, No. 6, 1191--1244 (2014; Zbl 1358.37046)]. The focus lies on the situation for which the given vector field, denoted by \(\mathbf b\), is not regular and even defined only up to \(\mathcal{L}^1 \times \mathcal{L}^d\) negligible sets. A regular flow is a solution of the initial value problem that is absolutely continuous and satisfies an additional inequality between certain measures related to a non-concentration property of trajectories. The existence of a unique regular Lagrangian flow is obtained under a regularity and a global growth condition for \(\mathbf b\). This solution is also stable with respect to smooth perturbations of \(\mathbf b\). Under weaker assumptions a regular generalized flow involving a kind of weak formulation with probability measures exists. A crucial step in the proof of the latter result is to show the existence of a distributional solution of the continuity equation \(w_t+\text{div} (\mathbf bw)=0\) using a mollifying technique. Applications are given to the Keyfitz-Kranzer system and the semi-geostrophic system. In two further sections the question of differentiability and a local theory with applications to the Vlasov-Poisson system are considered. Finally, the extension of the theory to metric measure spaces is discussed.