This paper discusses the Neumann–Neumann–Neumann–Dirichlet initial-boundary value problem with complicated nonlinear diffusion term, precisely, an extended May-Nowak-fluid system described by \(\begin{aligned} {\left\{ \begin{array}{ll} r_{t}+u\cdot \nabla r=\nabla \cdot (D(r)\nabla r)-\chi \nabla \cdot (r\nabla s)-r-rz+\varphi ,& \quad x\in \Omega ,t>0,\\ s_{t}+u\cdot \nabla s=\Delta s-s+rz,& \quad x\in \Omega ,t>0,\\ u\cdot \nabla z=\Delta z-z+s,& \quad x\in \Omega ,t>0,\\ u_{t}+\nabla P=\Delta u+r\nabla \phi ,& \quad x\in \Omega ,t>0,\\ \nabla \cdot u=0,& \quad x\in \Omega ,t>0, \quad \quad \quad \quad \quad \quad \quad \quad (*) \end{array}\right. } \end{aligned}\) where \(\Omega \subset \mathbb {R}^{3}\) is a smoothly bounded domain, and where \(D(r)\ge k_{D}r^{m-1}(k_{D}>0)\) . Provided \(m>\frac{14}{9}\) , then for any nonnegative initial data \((r_{0},s_{0},u_{0})\) , it is confirmed in the current work that at least one global weak solution to the corresponding problem (*) exists, and remains uniformly bounded. To the best of our knowledge, none of the results available so far seems applicable to such three-dimensional fluid-coupling system, which means that indeed, our reasoning represents a first step toward answering the question of how cross-diffusion, nonlinear diffusion and fluid mechanism interact, therefore supplementing the research in this direction.
