Axisymmetric vector field

A vector field on a rotation-invariant domain in $\mathbb R^3$ is axisymmetric about the $z$ -axis if

u(t,R_\phi x)=R_\phi u(t,x)

for every rotation $R_\phi$ about that axis. Away from the axis this is equivalent to writing

u=u_r(t,r,z)e_r+u_\theta(t,r,z)e_\theta+u_z(t,r,z)e_z

with all three cylindrical components independent of $\theta$ . The basis vectors $e_r,e_\theta$ themselves still depend on $\theta$ .

Scalar fields and incompressibility

A scalar is axisymmetric when $a(t,R_\phi x)=a(t,x)$ . For a differentiable axisymmetric velocity, incompressibility becomes

\frac1r\partial_r(ru_r)+\partial_z u_z=0\qquad(r>0).

The symmetry axis

The cylindrical description is singular at $r=0$ . Cartesian regularity at the axis imposes additional behavior on the components. Axisymmetry permits a nonzero azimuthal component; it does not mean absence of swirl.

References

Ben Pineau, Notes for Beale–Kato–Majda Blowup Criterion and Some Applications, axisymmetric Euler section.