Global classical solutions of 2D Keller–Segel–Navier–Stokes equations with logistic source

In this paper, we investigate the Keller–Segel system coupled with the Navier–Stokes equations in a bounded domain \(\Omega \subset \mathbb {R}^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ as follows \(\begin{aligned} \left\{ \begin{aligned}&n_{t}+u\cdot \nabla n=\Delta n-\nabla \cdot (n\nabla c)-\kappa n^{2},\\&c_{t}+u\cdot \nabla c=\Delta c -nc,\\&v_{t}+u\cdot \nabla v=\Delta v -\gamma v+n,\\&u_{t}+(u\cdot \nabla ) u+\nabla \pi =\Delta u-nf,\\&\nabla \cdot u=0. \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& n_t+u·\mathrm{\nabla }n=\mathrm{\Delta }n-\mathrm{\nabla }·(n\mathrm{\nabla }c)-\kappa n^2,\hfill \\ \hfill & c_t+u·\mathrm{\nabla }c=\mathrm{\Delta }c-nc,\hfill \\ \hfill & v_t+u·\mathrm{\nabla }v=\mathrm{\Delta }v-\gamma v+n,\hfill \\ \hfill & u_t+(u·\mathrm{\nabla })u+\mathrm{\nabla }\pi =\mathrm{\Delta }u-nf,\hfill \\ \hfill & \mathrm{\nabla }·u=0.\hfill \end{array}\right)\end{array}$$ We establish the global existence of classical solutions for the initial-boundary value problem in a bounded domain under mild assumption on the initial data.