A three-dimensional Keller-Segel-Navier–Stokes system involving subquadratic logistic degradation: global generalized solutions and eventual smoothness

In this paper, we consider a Keller-Segel-Navier–Stokes system involving subquadratic logistic degradation: $$\begin{aligned} \left\{ \begin{array}{ccl} n_t + {\textbf{u}}\cdot \nabla n & =& \Delta n-\nabla \cdot (n\nabla c)+\rho n- \mu n^{\alpha }, \\ \, c_t + {\textbf{u}}\cdot \nabla c & =& \Delta c -c+n, \\ {\textbf{u}}_t+ ({\textbf{u}}\cdot \nabla ) {\textbf{u}}& =& \Delta {\textbf{u}}+ \nabla P + n\nabla \phi , \\ \nabla \cdot {\textbf{u}}& = & 0 \end{array} \right. \end{aligned}$$ in a three-dimensional smoothly bounded domain along with reasonably mild initial conditions and no-flux/no-flux/Dirichlet boundary conditions, where $$\rho \in {\mathbb {R}}$$ and $$\mu >0$$ . The purpose of the present work is to firstly establish the generalized solvability for the model under the subquadratic exponent restriction $$\alpha \ge \frac{4}{3}$$ , which indicates that persistent Dirac-type singularities can be ruled out, and to secondly exhibit the eventual smoothness of these solutions under the stronger restriction $$\alpha > \frac{5}{3}$$ whenever $$\rho $$ is not too large in the sense of $$\begin{aligned} \big (\rho _++1\big )^{\alpha -1}{\rho _+}\le \delta _0\mu ^{\alpha },\qquad \big (\rho _++1\big )^{\min \{1,\,\alpha -1\}}{\rho _+}^{\max \{1,\,3-\alpha \}}\le \delta _0\mu ^{2},\qquad \rho _+\le \delta _0\mu \end{aligned}$$ for some $$\delta _0=\delta _0(\alpha )>0$$ . These results especially extend the precedent works due to Winkler (J Functional Anal 276: 1339-1401, 2019; Comm Math Phys 367: 439–489, 2022), where, among other things, the corresponding studies focus on the case $$\alpha =2$$ of quadratic degradation.