论文标题
集体行为的无压力流体动力模型的基础状态的全球时间稳定性
Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour
论文作者
论文摘要
我们考虑了一个无压的集体行为流体动力学模型，该模型与圆环上的密度函数$ρ$和速度场$ v $有关，并且由$ρ$，$ \ partial_tρ + \ m athrm {div}（div}（vρ）（vp）和$ v $ $ v $ v的连续性方程式描述\ nabla v -ΔV=-ρ\ nabla kρ$，具有与密度$ρ$相关的强迫建模集体行为，其中$ k $具有相互作用的潜力，定义为在$ \ mathbb {t}^d $上使用$ \ mathbb {t}^d $的poisson方程的解决方案。如果扰动$（ρ_0-1，v_0）$满足$ \ | v_0 \ | _ {b^{d/p-1} _ {p，1}（\ mathbb {t}^d）}} + \ | ρ_0-1\ | _ {b^{d/p} _ {p，1}（\ mathbb {t}^d）} \leqε$，用于足够小$ε> 0 $。
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 $ε>0$.
