  We show that for any given solenoidal initial data in $L^2$ and any
solenoidal external force in $L_{\text{loc}}^q \bigcap L^{3/2}$ with $q>3$,
there exist partially regular weak solutions of the Navier-Stokes equations in
$\R^4 \times [0,\infty[$ which satisfy certain local energy inequalities and
whose singular sets have locally finite $2$-dimensional parabolic Hausdorff
measure. With the help of a parabolic concentration-compactness theorem we are
able to overcome the possible lack of compactness arising in the spatially
$4$-dimensional setting by using defect measures, which we then incorporate
into the partial regularity theory.
