WELL-POSEDNESS AND ASYMPTOTIC STABILITY OF SOLUTIONS FOR THE INCOMPRESSIBLE TONER-TU MODEL

Abstract

In this paper, we investigate the Toner-Tu model describing the universal scaling of fluctuations in polar phases of dry active matter. The momentum equations are the incompressible Navier-Stokes-type equations containing the Rayleigh-Helmholtz friction term alpha v - beta|v|(2)v. When alpha < 0 and beta > 0, the fluid is damped to the disordered phase v = 0. By contrast, for alpha > 0 and beta > 0, v is pushed towards a nonvanishing velocity of magnitude root alpha/root beta corresponding to the ordered phase. In this paper, we deal with both cases with initial data in H-2(R-d). (1) For the ordered case, we perturb v around v0 of magnitude root alpha/root beta by defining u = v - v(0). By taking initial data u(0 )is an element of H-2(R-d) with small L-2(R-d) norm, we show that there exists a unique solution u globally in time and u decays algebraically, which verifies that v converges to v(0) as t -> infinity. (2) For the disordered case, we show that v exists uniquely global in time and decays exponentially. We further investigate the ordered phase case (with alpha = beta = 1) in polar coordinates to see the stability of a velocity vector of magnitude 1.