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

An analytical and computational study of the incompressible Toner-Tu Equations