We consider the following chemotaxis-fluid system in a two-dimensional, open, bounded domain \(\Omega \) : \(\begin{aligned} {\left\{ \begin{array}{ll} n_t + u \cdot \nabla n & = \Delta n - \chi \nabla \cdot \bigl (n c^\alpha \nabla c\bigr ) + r n - \mu n^2, \\ c_t + u \cdot \nabla c & = \Delta c - c + n, \\ u_t + u \cdot \nabla u & = \Delta u - \nabla P + n \nabla \phi , \\ \nabla \cdot u & = 0, \\ \end{array}\right. } \end{aligned}\) where \(r, \mu , \alpha , \chi \) are positive parameters and \(\phi \in W^{1,\infty }(\Omega )\) . In this paper, we show that the quadratic logistic source ensures the existence of global bounded solutions under the condition \(\alpha \in \left( 0, \frac{1}{2}\right) \) . Our proof relies on the Moser–Trudinger inequality, a parabolic regularity result in Orlicz spaces, and a variational approach.
