  In this paper we consider a parabolic toy-model for the incompressible
Navier-Stokes system. This model, as we shall see below, shares a lot of
similar features with the incompressible model; among which the energy
inequality, the scaling symmetry, and it is also supercritical in $3$D. Our
goal is to establish some regularity results for this toy-model in order to
get, if possible, better insight to the standard Navier-Stokes system. We also
prove here, in a direct manner, a Caffarelli-Kohn-Nirenberg type result for our
model. Finally, taking advantage of the absence of the divergence-free
constraint, we are able to study this model in the radially symmetric setting
for which we are able to establish full regularity.
