The impression I got form the equation is that you model your system as being composed of a collection of point particles. The index $i$ indexes this set of point particles (as opposed to indexing the components of a vector). 

One could create a spatial velocity distribution by saying $\vec{v}(\vec{r})=\sum_i \vec{v}_i \delta(\vec{r} - \vec{r}_i)$, where $\vec{v}_i$ is the velocity of the $i$th particle and $\vec{r}_i$ is the position of the $i$th particle. Then the spatial fourier transform of this $\vec{v}(\vec{r})$ is $\vec{v}(\vec{k})$ or, as you have written it, $\vec{v}_\vec{k}$.

That being said, it is unclear how to get a dispersion relation from this.