Nonlinear stability for active suspensions
Résumé
This paper is devoted to the nonlinear analysis of a kinetic model
introduced by Saintillan and Shelley
to describe suspensions
of active rodlike particles in viscous flows. We investigate the
stability of the constant state $\Psi(t,x,p) = \frac{1}{4\pi} $
corresponding to a distribution of particles that is homogeneous in
space (variable $x \in \mathbb{T}^3$) and uniform in orientation (variable
$p \in \mathbb{S}^2$). We prove its nonlinear stability under the optimal
condition of linearized spectral stability, without any addition of
spatial diffusion. The mathematical novelty and difficulty compared
to previous linear studies comes from the presence of a quasilinear
term in $x$ due to nonlinear convection. A key feature of our work,
which we hope to be of independent interest, is an analysis of
enhanced dissipation and mixing properties of the advection
diffusion operator
$$\partial_t + (p + u(t,x)) \cdot \nabla_x - \nu \Delta_p $$
on $\mathbb{T}^3 \times \mathbb{S}^2$ for a given appropriately small vector
field $u$.
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