On a toy-model related to the Navier–Stokes equations

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.