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

Abstract

We consider the incompressible Toner-Tu (ITT) partial differential equations (PDEs), which are an important example of a set of active-fluid PDEs. They share many properties with the Navier-Stokes equations (NSEs) but there are also important differences. The NSEs are usually considered either in the decaying or the additively forced cases, whereas the ITT equations have no additive forcing, but instead have a linear, activity term αu (with u the velocity field), which pumps energy into the system; they also have a negative u|u| 2 -term that stabilizes growth and provides a platform for either frozen or statistically steady states. These differences make the ITT equations an intriguing candidate for study using a combination of PDE analysis and pseudospectral direct numerical simulations (DNSs). In the d = 2 case, we have established global regularity of solutions. 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. For d = 3 there are equivalent bounded hierarchies for Leray-type weak solutions. We present results for these norms from our DNSs in both d = 2 and d = 3, and contrast them with their counterparts for the d = 3 NSEs.