This paper deals with a two-competing-species chemotaxis-fluid system with two different signals$$\begin{aligned} \left\{ \begin{aligned}&(n_{1})_{t}+\mathbf{u }\cdot \nabla n_{1}=d_{1}\Delta n_{1}-\chi _{1}\nabla \cdot (n_{1}\nabla c) +\mu _{1} n_{1}(1-n_{1}-a_{1}n_{2}),&\text {in}\; \Omega \times (0,\infty ), \\&c_{t}+\mathbf{u }\cdot \nabla c=d_{2}\Delta c-\alpha _{1} cn_{2},&\text {in}\; \Omega \times (0,\infty ), \\&(n_{2})_{t}+\mathbf{u }\cdot \nabla n_{2}=d_{3}\Delta n_{2}-\chi _{2}\nabla \cdot (n_{2}\nabla v) +\mu _{2}n_{2}(1-a_{2}n_{1}-n_{2}),&\text {in}\; \Omega \times (0,\infty ), \\&v_{t}+\mathbf{u }\cdot \nabla v=d_{4}\Delta v-\alpha _{2}vn_{1},&\text {in}\; \Omega \times (0,\infty ), \\&\mathbf{u }_{t}+\kappa (\mathbf{u }\cdot \nabla )\mathbf{u }=\Delta \mathbf{u } +\nabla P+(\beta _{1} n_{1}+\beta _{2} n_{2})\nabla \phi ,&\text {in}\; \Omega \times (0,\infty ), \\&\nabla \cdot \mathbf{u }=0,&\text {in}\; \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^{N}\), \(N=2,3\), under homogeneous Neumann boundary conditions for \(n_{1}, n_{2}, c, v\) and zero Dirichlet boundary condition for \(\mathbf{u }\), where \(\kappa \in \{0,1\}\), the parameters \(d_{i}\) (\(i=1,2,3,4\)) and \(\chi _{j},\mu _{j}, a_{j}, \alpha _{j},\beta _{j}\) (\(j=1,2\)) are positive. This system describes the evolution of two-competing species which react on two different chemical signals in a liquid surrounding environment. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. Firstly, when \(N=2\) and \(\kappa =1\), based on the standard heat-semigroup argument, it is proved that for all appropriately regular nonnegative initial data and any positive parameters, this system possesses a unique global bounded solution. Secondly, when \(N=3\) and \(\kappa =0\), by using the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that there exists \(\theta _{0}>0\) such that \(\frac{\max \{\chi _{1},\chi _{2}\}}{\min \{\mu _{1},\mu _{2}\}}<\theta _{0}\). Finally, by means of energy functionals and comparison arguments, it is shown that the global bounded solution of the system converges to different constant steady states, according to the different values of \(a_{1}\) and \(a_{2}\). Furthermore, we give the precise convergence rates of global solutions.

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具有两种化学物质的两个竞争物种趋化流体系统的全局有界性和稳定性

本文研究具有两个不同信号的具有两种竞争性的趋化性流体系统$$ \ begin {aligned} \ left \ {\ begin {aligned}＆（n_ {1}）_ {t} + \ mathbf {u} \ cdot \ nabla n_ {1} = d_ {1} \ Delta n_ { 1}-\ chi _ {1} \ nabla \ cdot（n_ {1} \ nabla c）+ \ mu _ {1} n_ {1}（1-n_ {1} -a_ {1} n_ {2}） ，＆\ text {in} \; \ Omega \ times（0，\ infty），\\＆c_ {t} + \ mathbf {u} \ cdot \ nabla c = d_ {2} \ Delta c- \ alpha _ {1} cn_ {2}，＆\文字{in} \; \ Omega \ times（0，\ infty），\\＆（n_ {2}）_ {t} + \ mathbf {u} \ cdot \ nabla n_ {2} = d_ {3} \ Delta n_ {2}- \ chi _ {2} \ nabla \ cdot（n_ {2} \ nabla v）+ \ mu _ {2} n_ {2}（1-a_ {2} n_ {1} -n_ {2}），＆\文字{in} \; \ Omega \ times（0，\ infty），\\＆v_ {t} + \ mathbf {u} \ cdot \ nabla v = d_ {4} \ Delta v- \ alpha _ {2} vn_ {1}，＆\文字{in} \; \ Omega \ times（0，\ infty），\\＆\ mathbf {u} _ {t} + \ kappa（\ mathbf {u} \ cdot \ nabla）\ mathbf {u} = \ Delta \ mathbf {u} + \ nabla P +（\ beta _ {1} n_ {1} + \ beta _ {2} n_ {2}）\ nabla \ phi，＆\ text {in} \; \ Omega \ times（0，\ infty），\\＆\ nabla \ cdot \ mathbf {u} = 0，＆\ text {in} \; \ Omega \ times（0，\ infty），\ end {aligned} \ right。\ end {aligned} $$在光滑的有界域\（\ Omega \ subset {\ mathbb {R}} ^ {N} \），\（N = 2,3 \）中，在\（n_ {1}，n_ { 2}，c，v \）和\（\ mathbf {u} \）的零Dirichlet边界条件，其中\（\ kappa \ in \ {0,1 \} \），参数\（d_ {i} \ ）（\（i = 1,2,3,4 \））和\（\ chi _ {j}，\ mu _ {j}，a_ {j}，\ alpha _ {j}，\ beta _ {j } \）（\（j = 1,2 \））是肯定的。该系统描述了在液体周围环境中对两种不同化学信号反应的两种竞争物质的进化。细胞和化学物质在由于细胞聚集而产生的力的作用下，通过粘性不可压缩的流体运输。首先，当\（N = 2 \）和\（\ kappa = 1 \）时，基于标准的热半群论证，证明了对于所有适当正则的非负初始数据和任何正参数，该系统都具有唯一的全局有界解。其次，当\（N = 3 \）和\（\ kappa = 0 \），通过使用最大Sobolev正则性和半群技术，证明了只要存在\（\ theta _ {0}> 0 \）使得\（\ frac {\ max \ {\ chi _ {1}，\ chi _ {2} \}} {\ min \ {\ mu _ {1}，\ mu _ {2} \}} <\ theta _ {0} \）。最后，通过能量泛函和比较参数，表明根据\（a_ {1} \）和\（a_ {2}的不同值，系统的全局有界解收敛到不同的恒定稳态。\）。此外，我们给出了全球解决方案的精确收敛速度。