Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour

Abstract

We consider a pressureless hydrodynamic model of collective behaviour, which is concerned with a density function $\rho$ and a velocity field $v$ on the torus, and is described by the continuity equation for $\rho$, ${\partial }_t\rho +\mathrm{div}(v\rho )=0$, and a compressible hydrodynamic equation for $v$, $\rho v_t+\rho v\cdot \nabla v-\Delta v=-\rho \nabla K\rho$ with a forcing modelling collective behaviour related to the density $\rho$, where $K$ stands for the interaction potential, defined as the solution to the Poisson equation on ${\mathbb{T}}^d$. We show global-in-time stability of the ground state $(\rho ,v)=(1,0)$ if the perturbation $({\rho }_0-1,v_0)$ satisfies $\Vert v_0{\Vert }_{B_{p,1}^{d/p-1}({\mathbb{T}}^d)}+\Vert {\rho }_0-1{\Vert }_{B_{p,1}^{d/p}({\mathbb{T}}^d)}\le \epsilon$ for sufficiently small $\epsilon >0$.

21 pages