An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation

Abstract

This paper focuses on the following Keller–Segel–Navier–Stokes system with rotational flux: in a bounded domain $$\Omega \subset {\mathbb {R}}^3$$ with a smooth boundary, where $$\kappa \in {\mathbb {R}}$$ is a given constant, $$\phi \in W^{1,\infty }(\Omega )$$ , $$|S(x,n,c)|\le C_S(1+n)^{-\alpha }$$ , and the parameter $$\alpha \ge 0$$ . If $$\alpha >\frac{1}{3}$$ , then, for all reasonable regular initial data, a corresponding initial-boundary value problem for (KSNF) possesses a globally defined weak solution. This result improves upon the result of Wang (Math Models Methods Appl Sci 27(14):2745–2780, 2017), in which the global very weak solution for the system (KSNF) is obtained. In comparison with the result of the corresponding fluid-free system, the optimal condition on the parameter $$\alpha $$ for global (weak) existence is established. Our proofs rely on a variant of the natural gradient-like energy functional.