A Lagrangian flow of a time-dependent vector field 
is a map 
whose trajectories solve
 |
with 
.
Thus 
labels the initial position and 
follows the trajectory through it. For a smooth
vector field, the trajectories are unique on their
intervals of existence. The formulation also applies to less regular vector
fields by requiring absolutely continuous
trajectories that solve the differential
equation for almost every time (DiPerna and Lions 1989, Ambrosio 2004).
A regular Lagrangian flow additionally controls the compression of Lebesgue measure. Such flows provide a formulation of transport by nonsmooth vector fields, including the velocity fields in Euler's equations of inviscid motion.
