Calculus of Variations and Geometric Measure Theory

E. Paolini - E. Stepanov

Flows of measures generated by vector fields

created by stepanov on 15 Jan 2014
modified by paolini on 21 Feb 2018


Published Paper

Inserted: 15 jan 2014
Last Updated: 21 feb 2018

Journal: Proc. Royal Soc. Edinburgh Ser. A - Mathematics
Year: 2018
Doi: 10.1017/S0308210517000312


We show that for a large class of measurable vector fields in the sense of N. Weaver (i.e. derivations over the algebra of Lipschitz functions), the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ``flows along'' the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

Keywords: measurable vector field, continuity equation, flow of measures, integral curve