【作者】 李锋；

【导师】 李玉祥；

【作者基本信息】 东南大学 ， 应用数学， 2020， 博士

【摘要】 本文研究了几类描述细胞在自身分泌的化学物质刺激下进行趋向性运动的偏微分方程组,包含趋化-趋触模型和两类趋化流体模型.论文研究了这几类趋化模型解的适定性,有界性以及带流体时的小对流极限问题.本文主要内容包含四部分:第一章概述了所研究趋化模型的生物背景,国内外的发展现状并简要介绍本文的主要工作.在第二章中,我们研究了具有一般logistic项的拟线性趋化-趋触模型(?)的Neumann边值问题,其中Ω(?)RN(N≥ 2)为光滑有界区域,χ>0,ζ>0,μ>0和r>1.扩散系数D(u)满足D(u)≥CDum-1,u>0,其中CD.通过利用Lp估计,Moser-Alikakos迭代方法以及连续性定理我们证明了当非线性扩散与logistic项之间的指数满足(?)时,则该问题存在整体有界的解.本章得到的结果推广和改进了部分已知结果.第三章考虑带旋转流的拟线性趋化流体模型(?)其中n和c满足Neumann边界条件,u满足Dirichlet边界条件.Ω(?)R3是一个具有光滑边界的有界区域,φ∈W1,∞(Ω),矩阵函数S(x,n,c)满足限制条件|S(x,n,c)| ≤S0(n+1)-α(α>0).本章证明了对于满足一定正则性的初值函数(n0,c0,u0),当m+α>4/3且m>1/3时,相应的初边值问题存在一个整体广义弱解.本章得到的结果推广和改进了部分已知结果.第四章我们考虑珊瑚受精模型(?)其中n,c,m满足Neumann边界条件,u满足Dirichlet边界条件.Ω(?)R2是一个具有光滑边界的有界区域.假设(nk,ck,mk,uk)是k∈(-1,1)对应的古典解且初值函数(n0,c0,m0,u0)满足一定正则性条件,则(nk,ck,mk,uk)关于k线性收敛到(n0,c0,m0,u0)并且收敛系数与时间有关.更多还原

【Abstract】 The present thesis studies several chemotaxis models in biomathematics,which describe the motion of cells,who,besides random diffusion,bias their movement towards a chemically more favorable environment,including the chemotaxis-hapotaxis model and two types of chemotaxis-fluid models.This thesis is devoted to studying the well-posedness,global boundedness and the small-convection limit for these chemotaxis models.The dissertation is divided into four parts.Chapter 1 gives a summary to the chemotaxis models involving the study background and our main results.In Chapter 2,a quasi-linear chemotaxis-hapotaxis model with logistic source(?)is considered under homogeneous Neumann boundary conditions in a smooth bounded domainΩ(?)RN,where χ>0,ζ>0,μ>0 and r>1.The diffusivity D(u)stasifies D(u)≥ CDum-1 for all u>0 with some constant CD>0.By application of the Lp esimate,Moser-Alikakos iteration mehtod and continuity argument,we have proved that the indexes in nonlinear diffusion and the logistic term,which satisfies(?)have guaranteed the global boundedness of classical solutions.This result generalize and improve some previously known ones.In Chapter 3,a chemotaxis-fluid system with nonlinear diffusion and rotational flux given by(?)is considered under omogeneous boundary conditions of Neumann type for n and c,and of Dirichlet type for u,where Ω(?)R3 is bounded domain with smooth boundary condition,φ∈W1,∞(Ω),the tensor-valued function S(x,n,c)is dominated by |S(x,n,c)| ≤ S0(n+1)-α(α>0).For all sufficiently smooth initial data(n0,c0,u0),if m+α>4/3 and m>1/3,the corresponding initial-boundary value problem possesses at least one global generalized weak solution.This result generalize and improve some previously known ones..In chapter 4,a four-component chemotaxis-fluid system(?)is considered,which is used to model coral fertilization.Here,n,c,and m satisfy hemogenous Neumann boundary,u fulfills homogeneous Dirichlet boundary condition,Ω(?)R2 is a bounded domain with smooth bouandry.For any k ∈(-1,1),assume that(nk,ck,mk,uk)is the corre-sponding global classical solution and the initial data is smooth enough,then(nk,ck,mk,uk)will stabilize to(n0,c0,m0,u0)with an explicit rate and a time dependent coefficient as k→ 0.更多还原