二维趋化-流体耦合模型解的稳定性分析  

Stability Analysis of the Solution to a Two-dimensional Chemotaxis-fluid Coupled Model

作　　者：李嘉文 刘梦琦 侯智博 LI Jiawen;LIU Mengqi;HOU Zhibo(School of Science,Xihua University,Chengdu 610039,Sichuan)

机构地区：[1]西华大学理学院,四川成都610039

出　　处：《四川师范大学学报（自然科学版）》2023年第3期352-361,共10页Journal of Sichuan Normal University（Natural Science）

基　　金：四川省教育厅自然科学基金重点项目(17ZA0357);四川省科技厅自然科学基金项目(2018JY0503)。

摘　　要：考虑在二维有界凸区域Ω上具有信号产生机制的chemotaxis-Navier-Stokes方程组{n_(t)+u·■n=△n-■·(nS(n)■c),x∈Ω,t>0,c_(t)+u·■c=△c-c+n,x∈Ω,t>0,u_(t)+(u·■)u+■P=△u+n■Ф,x∈Ω,t>0,■·u=0,x∈Ω,t>0的初边值问题,其中n和c满足齐次Neumann边界条件,u满足Dirichlet边界条件;S(n)和Φ是给定的足够光滑的函数,且满足|S(x,n,c)|≤C_(S)(1+n)^(-α).此前的研究结果表明:该模型在二维有界凸区域中存在整体有界经典解.进一步研究了此模型解的大时间渐近行为:在α>0,C_(S)足够小的情况下,当时间t趋于无穷时,模型的经典解以指数形式收敛到常数平衡态(n_(0),n_(0),0),其中n_(0)=1/|Ω|∫_(Ω) n_(0),dx,n_(0)=n(x,0).The chemotaxis-Navier-Stokes system with signal production in two-dimensional bounded convex domain Ω,as given by {n_(t)+u·■n=△n-■·(nS(n)■c),x∈Ω,t>0,c_(t)+u·■c=△c-c+n,x∈Ω,t>0,u_(t)+(u·■)u+■P=△u+n■Ф,x∈Ω,t>0,■·u=0,x∈Ω,t>0,is considered under homogeneous boundary conditions of Neumann type for n and c,and of Dirichlet type for u.Here S(n),Ф are known sufficiently smooth functions,while S(x,n,c)satisfies |S(x,n,c)|≤C_(S)(1+n)^(-α).It has been known that this model possesses a global bounded classical solution in two-dimensional bounded convex domain.This paper further studies the large time asymptotic behavior of the solution within the model.If α>0 and C_(S)is properly small,the classical solution of the model approachs to a constant equilibrium state (n_(0),n_(0),0)exponentially when time t is infinite,where n_(0)=1/|Ω|∫_(Ω) n_(0),dx,n_(0)=n(x,0).

关 键 词：趋化-流体耦合模型 有界性 稳定性 

分 类 号：O175.29[理学—数学]