# 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*] × ℝ^{d} → ℝ^{d} be a bounded Borel vector field,
*T* > 0 and let \bar μ be a non-negative Radon measure on ℝ^{d}. 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.

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