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

TL;DR

This paper analyzes the incompressible Toner-Tu equations, combining PDE analysis and numerical simulations to explore their properties, solutions, and differences from Navier-Stokes equations in 2D and 3D.

Contribution

It provides the first global regularity results for 2D solutions and establishes bounded hierarchies of norms for both 2D and 3D cases, with numerical comparisons to Navier-Stokes.

Findings

Global regularity of 2D solutions established.

Bounded hierarchies of norms for higher derivatives and moments.

Numerical simulations contrast ITT properties with Navier-Stokes equations.

Abstract

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.