  The incompressible Toner-Tu (ITT) partial differential equations (PDEs) are
an important example of a set of active-fluid PDEs. While they share certain
properties with the Navier-Stokes equations (NSEs), such as the same scaling
invariance, there are also important differences. The NSEs are usually
considered in either the decaying or the additively forced cases, whereas the
ITT equations have no additive forcing. Instead, they include a linear,
activity term $\alpha \bu$ ($\bu$ is the velocity field) which pumps energy
into the system, but also a negative $\bu|\bu|^{2}$-term which provides a
platform for either frozen or statistically steady states. Taken together,
these differences make the ITT equations an intriguing candidate for study
using a combination of PDE analysis and pseudo-spectral direct numerical
simulations (DNSs). In the $d=2$ case, we have established global regularity of
solutions, but we have also shown the existence of bounded hierarchies of
weighted, time-averaged norms of both higher derivatives and higher moments of
the velocity field. Similar bounded hierarchies for Leray-type weak solutions
have also been established in the $d=3$ case. We present results for these
norms from our DNSs in both $d=2$ and $d=3$, and contrast them with their
Navier-Stokes counterparts.
