Global boundedness in a 2D Chemotaxis-Navier-Stokes system with density-dependent motion

This paper establishes the global existence of classical solutions to an initial-boundary value problem for a coupled Keller-Segel-Navier-Stokes system in two-dimensional. The model is governed by the following equations$ \begin{equation*} \left\{\begin{aligned} & n_t = \Delta(\gamma(c)n)+\mu n(1-n^{\sigma})-\mathbf{u}\cdot\nabla n, && x\in\Omega, t>0,\\ & c_t = \Delta c-c+n^{1-\alpha}c^{\alpha}-\mathbf{u}\cdot\nabla c,&& x\in\Omega, t>0,\\ & \mathbf{u}_t = \Delta \mathbf{u}-\nabla P+n\nabla\phi+\kappa (\mathbf{u}\cdot\nabla)\mathbf{u},&& x\in\Omega, t>0,\\ & \nabla\cdot \mathbf{u} = 0, && x\in\Omega, t>0,\\ \end{aligned}\right. \end{equation*} $where $ \sigma>0, \alpha\in(0,1) $ are positive constants, $ \Omega\subset\mathbb{R}^2 $ denotes a bounded domain with a smooth boundary, and the function $ \gamma $ belongs to $ C^3([0,\infty)) $ with $ \gamma(s)>0 $ for all $ s\geq0 $, as well as the gravitational potential $ \phi\in W^{2,\infty}(\Omega) $. Under homogeneous Neumann boundary conditions and for any $ \kappa\in\mathbb{R} $, we establish the existence and uniqueness of global classical solution for this Keller-Segel-Navier-Stokes system.