I'm trying to solve a Stokes flow problem with a pseudo-spectral method in periodic boundary conditions.
The equations of interest are
$-\nabla^2 \bf{v} + \nabla p = \bf{f} \\ \nabla \cdot \bf{v} = 0$
where $\bf{f}$ is a force vector coming from the gradient of a stress tensor, $\bf{f} = \nabla \cdot \sigma$.
To do so, I use a transverse projection operator (i.e. the Green's function/Oseen tensor) in Fourier space as follows
$\bf{\tilde{v}} = \bf{\tilde{T}} \cdot \bf{\tilde{f}}$
where $\bf{\tilde{T}}$ is given by
$\bf{\tilde{T}} = \frac{1}{k^2} \left [ \bf{I} - \frac{\bf{k}\bf{k}}{k^2} \right ]$
My question is: What is the appropriate way to treat the zero k-mode? In the statement above the solution is undefined for $k = 0$. I think I should be able to obtain the volume averaged velocity from this mode, and I can't think off-hand why it shouldn't be well-defined. (Note this problem is similar to this related question).