{"title":"Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour","authors":"Piotr B. Mucha, Wojciech S. Ożański","doi":"10.4310/cms.2023.v21.n7.a9","DOIUrl":null,"url":null,"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 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 $\\epsilon>0$.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"110 1","pages":"0"},"PeriodicalIF":1.2000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Sciences","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/cms.2023.v21.n7.a9","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
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 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 $\epsilon>0$.
期刊介绍:
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