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

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n7.a9

Authors

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.)

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 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.

Keywords

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

2010 Mathematics Subject Classification

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

Received 3 November 2022

Accepted 25 January 2023

Published 9 October 2023