Particle trajectory of a velocity field

A particle trajectory for a velocity field $u$ , starting at $a$ at time $s$ , solves

\frac{d}{dt}X(t;s,a)=u(t,X(t;s,a)), \qquad X(s;s,a)=a.

Where these initial-value problems have unique solutions, $a\mapsto X(t;s,a)$ is the flow map from time $s$ to time $t$ .

Local existence

Continuity in time and local Lipschitz continuity in position give local existence and uniqueness by the Picard–Lindelöf theorem. The trajectory is followed only while it remains in the region and time interval where the field is defined.

Lagrangian description

The coordinate $a$ labels a particle. A field sampled along that trajectory is $q(t,X(t;s,a))$ ; its time derivative is the material derivative of $q$ .

References

Lenya Ryzhik, Lecture Notes for Math 256B (2024), Sections 1–3.