Moscow Mathematical Journal

Volume 18, Issue 1, January–March 2018  pp. 85–92.

A Necessary and Sufficient Condition for Existence of Measurable Flow of a Bounded Borel Vector Field

Authors:  Nikolay A. Gusev (1)
Author institution:(1) Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St, Moscow, 119991;
Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700;
RUDN University, 6 Miklukho-Maklay St, Moscow, 117198

Summary:

Let b: [0, T] × ℝd → ℝd be a bounded Borel vector field, T > 0 and let \bar μ be a non-negative Radon measure on ℝd. We prove that a \bar μ-measurable flow of b exists if and only if the corresponding continuity equation has a non-negative measure-valued solution with the initial condition \bar μ.

2010 Math. Subj. Class. 35D30, 34A12, 34A36.

Keywords: Continuity equation, non-smooth vector field, measure-valued solutions, flow, ordinary differential equation.