Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour
英文摘要：
We consider a pressureless hydrodynamic model of collective behaviour, which is concerned with a density function $ρ$ and a velocity field $v$ on the torus, and is described by the continuity equation for $ρ$, $\partial_t ρ+ \mathrm{div} (vρ)=0$, and a compressible hydrodynamic equation for $v$, $ρv_t + ρv\cdot \nabla v - Δv = -ρ\nabla K ρ$ with a forcing modelling collective behaviour related to the density $ρ$, 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 $(ρ, v)=(1,0)$ if the perturbation $(ρ_0-1 ,v_0)$ satisfies $\| v_0 \|_{B^{d/p-1}_{p,1}(\mathbb{T}^d )} + \| ρ_0-1 \|_{B^{d/p}_{p,1}(\mathbb{T}^d )} \leq ε$ for sufficiently small $ε>0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
