Communications in Mathematical Sciences

Volume 21 (2023)

Number 7

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

Pages: 1937 – 1959



Piotr B. Mucha (Institute of Applied Mathematics and Mechanics, University of Warsaw, Poland)

Wojciech S. Ożański (Department of Mathematics, Florida State University, Tallahassee, Fl., U.S.A.)


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 bythe continuity equation for $\rho, \rho_t + \mathrm{div}(v \rho) = 0$, 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 repulsive 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 ${\lVert v_0 \rVert}_{B^{d/p-1}_{p,1}(\Theta^d )} + {\lVert \rho_0-1 \rVert}_{B^{d/p}_{p,1}(\Theta^d )} \leq \epsilon$, where $p\in (\min (d/2,2),d)$ and $\epsilon >0$ is sufficiently small.


pressureless hydrodynamic model, stability, collective behaviour, Besov spaces, repulsive system

2010 Mathematics Subject Classification

35A01, 35A02, 35B20, 35B35, 35E15, 35K57, 35Q70, 35Q92

P.B.M. was supported by the Polish National Science Centre’s Grant No. 2018/30/M/ST1/00340 (HARMONIA), and W.S.O. was supported in part by the Simons Foundation.

Received 3 November 2022

Accepted 25 January 2023

Published 9 October 2023