Global boundedness of classical solutions to a Keller-Segel-Navier-Stokes system involving saturated sensitivity and indirect signal production in two dimensions

This paper is concerned with the following Keller–Segel–Navier–Stokes system with indirect signal production and tensor-valued sensitivity:

$ \left\{\begin{array}{*5{lllll }} n_{t}+u \cdot \nabla n=\Delta n-\nabla \cdot(n S(x,n,v,w) \nabla v), \quad &x \in \Omega, t>0, \\ v_{t}+u \cdot \nabla v=\Delta v-v+w, \quad &x \in \Omega, t>0, \\ w_{t}+u \cdot \nabla w=\Delta w-w+n, \quad &x \in \Omega, t>0, \\ u_{t}+\kappa(u \cdot \nabla) u+\nabla P=\Delta u+n \nabla \phi, \quad &x \in \Omega, t>0, \\ \nabla \cdot u=0, \quad &x \in \Omega, t>0, \end{array}\right. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (♡)$

in a bounded domain $ \Omega\subset \mathbb{R}^2 $ with smooth boundary, where $ \kappa \in \mathbb{R} $, $ \phi \in W^{2, \infty}(\Omega) $, and $ S $ is a given function with values in $ \mathbb{R}^{2\times2} $ which satisfies $ |S(x, v, w, u)|\leq C_{S}(n+1)^{-\alpha} $ with $ C_{S} > 0 $. If $ \alpha > 0 $, then for any sufficiently smooth initial data, there exists a globally classical solution which is bounded for the corresponding initial-boundary value problem of system (♡).