Nonlinear stability for active suspensions

Abstract

This paper is devoted to the nonlinear analysis of a kinetic model introduced by Saintillan and Shelley for 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. The main achievement in this work is that the smallness condition on the initial perturbation is independent of the translational diffusion and only depends on the rotational diffusion, which is particularly relevant for dilute suspensions. Upgrading our previous linear study [Ann. PDE 9 (2023), no. 2, article no. 20] to such a nonlinear stability result requires new mathematical ideas, due to the presence of a quasilinear term in $x$ associated with nonlinear convection. This term cannot be treated as a source, because it is not controllable by the rotational diffusion in $p$. Also, it prevents the decoupling of $x$-Fourier modes crucially used in Coti Zelati, Dietert, and Gérard-Varet [Ann. PDE 9 (2023), no. 2, article no. 20]. A key feature of our work 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 $u$. We hope this linear analysis is of independent interest, and useful in other contexts with partial or anisotropic diffusions.