In this paper, we consider the following system: $$\left\{\begin{array}{c}{n}_t+u\cdot \nabla n=\mathrm{\Delta }n-\nabla \cdot (n\chi (c)\nabla c),\hfill \\ c_t+u\cdot \nabla c=\mathrm{\Delta }c-cn,\hfill \\ u_t+\kappa (u\cdot \nabla )u=\mathrm{\Delta }u+\nabla P+n\nabla \mathrm{\Phi },\hfill \end{array}\right.$$ in a smoothly bounded domain $\mathrm{\Omega }\subset {ℝ}^2,$ with $\kappa \in \{0,1\}$ and a given function $$\chi (c)=\frac{1}{c^{𝜃}}$$ with $𝜃\in [0,1).$ It is proved that if $\kappa =1$ then for appropriately small initial data an associated no-flux/no-flux/Dirichlet initial-boundary value problem is globally solvable in the classical sense, and that if $\kappa =0$ then under a different but still suitable smallness restriction of the initial data, a corresponding initial-boundary value problem subject to no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded and approaches a constant equilibria $({\bar{n}}_0,0,0)$ in $L^{\infty }(\mathrm{\Omega })\times W^{1,\infty }(\mathrm{\Omega })\times L^{\infty }(\mathrm{\Omega })$ as $t\to \infty ,$ with ${\bar{n}}_0:=\frac{1}{\left|\mathrm{\Omega }\right|}{\int }_{\mathrm{\Omega }}{n}_0.$

在本文中，我们考虑以下系统：$$\left\{\begin{array}{c}{n}_t+你\cdot \nabla n=\mathrm{\Delta }n-\nabla \cdot （n\chi （C）\nabla C）,\hfill \\ C_t+你\cdot \nabla C=\mathrm{\Delta }C-Cn,\hfill \\ {你}_t+\kappa （你\cdot \nabla ）你=\mathrm{\Delta }你+\nabla 磷+n\nabla \mathrm{\Phi },\hfill \end{array}\right.$$在光滑有界域中$\mathrm{\Omega }\subset {ℝ}^2,$和$\kappa \epsilon \{0,1\}$和给定的函数$$\chi （C）=\frac{1}{C^{𝜃}}$$和$𝜃\epsilon [0,1）。$事实证明，如果$\kappa =1$那么对于适当小的初始数据，相关的无通量/无通量/狄利克雷初始边界值问题在经典意义上是全局可解的，并且如果$\kappa =0$然后，在初始数据的不同但仍然合适的小限制下，受无通量/无通量/狄利克雷边界条件影响的相应初始边值问题承认唯一的经典解，该解是全局有界的并且接近恒定平衡$（{\bar{n}}_0,0,0）$在$L^{\mathrm{无穷大}}（\mathrm{\Omega }）\times {瓦}^{1,\mathrm{无穷大}}（\mathrm{\Omega }）\times L^{\mathrm{无穷大}}（\mathrm{\Omega }）$作为$t\to \mathrm{无穷大},$和${\bar{n}}_0：=\frac{1}{\left|\mathrm{\Omega }\right|}{\int }_{\mathrm{\Omega }}{n}_0。$