Fluid velocity field

A fluid velocity field on a spatial domain $\Omega\subseteq\mathbb R^d$ and time interval $I$ is a map

u:I\times\Omega\to\mathbb R^d.

The vector $u(t,x)$ is the instantaneous velocity of fluid at the spatial point $x$ and time $t$ . This is the Eulerian description: the coordinate $x$ labels a location, rather than a fixed fluid particle.

Regularity and components

Regularity is an additional hypothesis; one may study smooth, continuous, or measurable velocity fields. The scalar components $u_i$ are taken in a specified basis. A time slice $u(t,\cdot)$ is a Euclidean vector field.

Particle motion is described by the trajectory equation $X'(t)=u(t,X(t))$ .

References

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