This work addresses the global solvability of a coupled chemotaxis-Stokes system with competitive Lotka–Volterra kinetics in a two-dimensional incompressible fluid medium. The system under study is formulated as: \(\begin{aligned} {\left\{ \begin{array}{ll} u_{1t}-w\cdot \nabla u_{1}=\Delta u_{1}-\chi \nabla \cdot \bigl (u_{1}\nabla v_{1}\bigl )+u_{1}\bigl (\lambda _{1}-\mu _{1}u_{1}^{r_{1}-1}+au_{2}\bigl ),& \quad x\in \Omega ,t>0,\\ u_{2t}-w\cdot \nabla u_{2}=\Delta u_{2}+\xi \nabla \cdot \bigl (u_{2}\nabla v_{2}\bigl )+u_{2}\bigl (\lambda _{2}-\mu _{2}u_{2}^{r_{2}-1}-bu_{1}\bigl ),& \quad x\in \Omega ,t>0,\\ v_{1t}-w\cdot \nabla v_{1}=\Delta v_{1}-v_{1}+u_{2},& \quad x\in \Omega ,t>0,\\ -w\cdot \nabla v_{2}=\Delta v_{2}-v_{2}+u_{1},& \quad x\in \Omega ,t>0,\\ w_{t}-\Delta w=(u_{1}+u_{2})\nabla \phi -\nabla P,& \quad x\in \Omega ,t>0,\\ \nabla \cdot w=0,& \quad x\in \Omega ,t>0\\ \end{array}\right. } \end{aligned}\) within a smoothly bounded domain \(\Omega \subset \mathbb {R}^2\) , we impose no-flux boundary conditions on \(u_1\) , \(u_2\) , \(v_1\) , and \(v_2\) , and no-slip boundary conditions on w, where \(\chi > 0\) , \(\xi > 0\) , \(a > 0\) , \(b > 0\) , \(\lambda _i > 0\) , \(\mu _i > 0\) for \(i = 1, 2\) and \( \phi \in W^{2,\infty }(\Omega ) \) . This coupled system represents the interaction between chemotaxis equations and the behavior of viscous incompressible fluids. In this study, we demonstrate the existence of a globally bounded classical solution to the associated problem under the assumptions \(r_{1}>1\) and \(r_{2}>1\) .
