This paper rigorously analyzes a Keller–Segel–Navier–Stokes system with indirect signal production and subquadratic logistic degradation: 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} n_t + {u} \cdot \nabla n = \Delta n - \chi \nabla \cdot ( n \nabla v )+\rho n-\mu n^\alpha , & x \in \Omega , \, t> 0, \\ v_t + {u} \cdot \nabla v = \Delta v - v + w, & x \in \Omega , \, t> 0, \\ w_t + {u} \cdot \nabla w = \Delta w - w + n, & x \in \Omega , \, t> 0, \\ {u}_t + ({u} \cdot \nabla ){u} + \nabla P = \Delta {u} + n \nabla \phi , & x \in \Omega , \, t> 0,\\ \nabla \cdot {u} = 0, & x \in \Omega , t > 0. \end{array}\right. } \end{aligned}\) On a bounded smooth domain \(\Omega \subset \mathbb {R}^3\) with no-flux for n, v, w and no-slip for u, parameters are: \(\rho \in \mathbb {R}\) , \(\mu > 0\) , \(\chi > 0\) (chemotactic sensitivity), \(\phi \in W^{2,\infty }(\Omega )\) , and \(\alpha \) (logistic exponent) critically affecting dynamics. A known bottleneck for weak degradation ( \(\alpha < 4/3\) ) is the need for an \(L^{4/3}\) estimate of n to obtain fluid energy bounds. While \(\alpha \ge 4/3\) yields solutions directly [10, 11, 43, 63, 68], the case \(\alpha < 4/3\) remains open in 3D. To address this, we construct a quasi-energy inequality: 0.2 \(\begin{aligned} \int _{\Omega } n^{\frac{2}{3}} + A \int _{\Omega } |\nabla v|^2 + B \int _{\Omega } |{u}|^2, \end{aligned}\) with large constants A, B. For \(\alpha > 5/4\) , we combine analysis of v’s regularity with the w-equation to obtain an \(L^{q}\) estimate ( \(q > 3/2\) ) for w, leading via Moser iteration to an \(L^{\infty }\) bound for v. Under small \(\chi \) , we prove uniform boundedness of this quasi-energy, overcoming classical barriers and providing new tools for such systems.
