Abstract

I will present a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles. This model, involving the distribution in space and orientation of the particles, is known to exhibit phase transitions. We focus on the linear analysis of incoherence, that is on the linearized equation around the uniform distribution, in the regime of parameters corresponding to spectral (neutral) stability. I will show that in the absence of rotational diffusion, the supension experiences a mixing phenomenon similar to Landau damping, and we provide optimal pointwise in time decay rates in weak topology. We show that this phenomenon persists for small rotational diffusion, and is combined with an enhanced dissipation at time scale $\nu^{-1/2}$. The mathematical interest of the model is that the usual velocity variable of kinetic models is replaced by an orientation variable on the sphere. The associated orientation mixing leads to limited algebraic decay for macroscopic quantities.

Joint work with Michele Coti Zelati and David Gérard-Varet