Regular Lagrangian Flow

A regular Lagrangian flow of a Borel measurable vector field is a measurable map satisfying two conditions. For almost every initial point with respect to Lebesgue measure, the trajectory is absolutely continuous, starts at , and satisfies for almost every . There is also a finite constant , independent of , such that

where is Lebesgue measure and denotes the pushforward measure (DiPerna and Lions 1989, Ambrosio 2004).

The inequality is bounded compressibility: for every measurable set , the measure of its preimage under is at most . A measure-preserving flow satisfies the stronger equality . Almost-everywhere invertibility is an additional property, rather than part of the definition. Regular Lagrangian flows extend the trajectory description of a Lagrangian flow to nonsmooth vector fields.