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

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 $\| v_0 \|_{B^{d/p-1}_{p,1}(\mathbb{T}^d )} + \| \rho_0-1 \|_{B^{d/p}_{p,1}(\mathbb{T}^d )} \leq \epsilon$ for sufficiently small $\epsilon>0$.