Cylindrical swirl diffusion operator

For an axisymmetric vector field $u=K(r,z)e_\theta$ , its vector Laplacian is

\Delta u=\left(K_{rr}+\frac1rK_r-\frac1{r^2}K+K_{zz}\right)e_\theta.

The radial swirl diffusion operator is therefore $L_\theta=\partial_r^2+r^{-1}\partial_r-r^{-2}$ , rather than the scalar radial Laplacian. The formula holds for $r>0$ .

Origin of the extra term

In Cartesian components, the angular basis satisfies $\partial_\theta^2e_\theta=-e_\theta$ , producing $-K/r^2$ . Writing $K=r\Omega$ gives

L_\theta(r\Omega)=r\left(\Omega_{rr}+\frac3r\Omega_r\right).

The bracket is the radial scalar Laplacian in four dimensions. Smoothness across the axis still requires the corresponding Cartesian regularity of the swirl field.