Global Weak Solutions in a Three-dimensional Keller–Segel–Navier–Stokes System with Flux Limitation and Superlinear Production

Abstract

This paper is concerned with a three-dimensional Keller–Segel–Navier–Stokes system incorporating singular flux limitation and superlinear production. The primary goal is to establish global existence of weak solutions under conditions ensuring that flux limitations suppress the blow-up tendencies induced by superlinear growth. More precisely, this paper focuses on the system

in a bounded domain \(\Omega \subset \mathbb {R}^3\) with smooth boundary, where \(0< \alpha < 1\) and \(\beta \ge 1\). Under the assumption \(\alpha > 1 - \frac{1}{3\beta -1}\), we prove global existence of weak solutions to the Neumann problem for \((*)\). This study extends the previous work by Winkler [27], in which the corresponding system with the regular sensitivity \((|\nabla c|^2+1)^{-\frac{\alpha }{2}}\) and the linear production \((\beta =1)\) was considered, and highlights how strong flux limitation can control the effects of superlinear growth.