Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier--Stokes system with logistic source

TL;DR

This paper proves the existence of global weak solutions for a 3D chemotaxis-Navier-Stokes system with degenerate diffusion and logistic growth, extending previous results to new parameter regimes.

Contribution

It establishes the first global weak solution existence results for the system with degenerate diffusion ($m>0$), filling a gap in the mathematical analysis of chemotaxis-fluid models.

Findings

Proved global existence of weak solutions for the system with degenerate diffusion.

Extended mathematical understanding of chemotaxis-Navier-Stokes systems with logistic source.

Built upon and generalized previous results for specific cases.

Abstract

This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t + u\cdot\nabla n =\Delta n^m - \chi\nabla\cdot(n\nabla c) + \kappa n -\mu n^2$, $x \in \Omega,\ t>0$, $c_t + u\cdot\nabla c = \Delta c - nc$, $x \in \Omega,\ t>0$, $u_t + (u\cdot\nabla)u = \Delta u + \nabla P + n\nabla\Phi, \quad \nabla\cdot u = 0$, $x \in \Omega,\ t>0$, where $\Omega\subset \mathbb{R}^3$ is a bounded domain and $\chi,\kappa \ge 0$ and $m, \mu >0$. In the above system without fluid environment Jin (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with $m=1$, Lankeit (Math.\ Models Methods Appl. Sci., 2016) established global existence of weak solutions. However, the above system with $m>0$ has not been studied yet. The purpose of this talk is to establish global existence of weak solutions in the chemotaxis-Navier--Stokes system with degenerate diffusion and logistic term.