| Abstract: |
| In this paper, we study a two-species chemotaxis-Navier-Stokes system with Lotka-Volterra type competitive kinetics: $n_t+u\cdot\nabla n=\Delta n-\chi_1\nabla\cdot(n\nabla w)+n(\lambda_1-\mu_1n^{\theta-1}-a_1v)$; $v_t+u\cdot\nabla v=\Delta v-\chi_2\nabla\cdot(v\nabla w)+v(\lambda_2-\mu_2v-a_2n)$; $w_t+u\cdot\nabla w=\Delta w-w+n+v$; $u_t+\kappa(u\cdot\nabla)u=\Delta u+\nabla P+(n+v)\nabla\phi$; $\nabla\cdot u=0$, $x\in \Omega$, $t>0$ in a bounded and smooth domain $\Omega\subset \mathbb{R}^2$ with no-flux/Dirichlet boundary conditions, where $\chi_1, \chi_2$ are positive constants. We present the global existence of generalized solution to a two-species chemotaxis-Navier-Stokes system and the eventual smoothness already occurs in systems with much weaker degradation $(\theta>1)$, again under a smallness condition on $\lambda_1, \lambda_2$. |
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