Global solvability in a 3D attraction-repulsion Navier-Stokes system with consumption of chemoattractant and sub-quadratic degradation

Abstract

This paper investigates an attraction-repulsion Navier-Stokes system that incorporates the consumption of chemoattractant and sub-quadratic degradation:$\begin{array}{c}\{\begin{array}{cc}{n}_t+u·\nabla n=\Delta n-\nabla ·\left(n\nabla c\right)+\nabla ·\left(n\nabla w\right)+f\left(n\right),& x\in \Omega ,t>0,\\ c_t+u·\nabla c=\Delta c-cn,& x\in \Omega ,t>0,\\ w_t+u·\nabla w=\Delta w-w+n,& x\in \Omega ,t>0,\\ u_t+(u·\nabla )u=\Delta u+\nabla P+n\nabla \Lambda ,\nabla ·u=0,& x\in \Omega ,t>0\end{array}\end{array}$in a bounded and smooth domain $\Omega \subset R^3$, with no-flux/Dirichlet boundary conditions and nonnegative integrable initial data, where f ∈ C¹([0, ∞)) is supposed to generalize standard choices of logistic-type reproduction and degradation, as obtained on letting $f\left(s\right)=\rho s-\mu s^{\alpha }$ for s ≥ 0, with ρ ≥ 0, μ > 0 and α ∈ (1, 2). In this work, it is demonstrated that, in addition to the fundamental condition that f(0) must be nonnegative, the assumption$\begin{array}{c}\frac{f\left(s\right)}{s}\to -\infty \text{as}s\to \infty \end{array}$which reflects the superlinearity of degradation, is sufficient to construct globally defined, nonnegative, and integrable functions n, c, w and u. These functions (n, c, w, u) provide a solution to the associated initial-boundary value problem in an appropriately generalized sense.