Affine velocity field

An affine velocity field on $\mathbb R^d$ has the form

u(t,x)=A(t)x+b(t),

with a matrix $A(t)$ and vector $b(t)$ . It is incompressible exactly when $\operatorname{tr}A(t)=0$ .

Exact momentum calculation

For differentiable $A,b$ ,

\partial_tu+(u\cdot\nabla)u =(A'+A^2)x+b'+Ab, \qquad \Delta u=0.

If $\operatorname{tr}A=0$ and $S=A'+A^2$ is symmetric, the unforced Euler and Navier–Stokes momentum equations are solved by

p(t,x)=-\tfrac12 x^{\mathsf T}S(t)x-(b'(t)+A(t)b(t))\cdot x,

up to an additive function of time. Conversely, on the whole space a pressure gradient cancelling this affine acceleration requires $A'+A^2$ to be symmetric.

Integrability

A nonzero affine velocity on $\mathbb R^d$ has infinite total kinetic energy. Such fields are still useful as local models or backgrounds; satisfying the local equations does not supply the decay required by a finite-energy whole-space problem.

References

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