Global existence and boundedness of chemotaxis-fluid equations to the coupled Solow-Swan model

In this paper, we consider the following Keller-Segel-(Navier)-Stokes system to the coupled Solow-Swan model

$$\{\begin{array}{rll}& n_t+u\cdot \mathrm{\nabla }n=\mathrm{\Delta }n-\chi \mathrm{\nabla }\cdot (n\mathrm{\nabla }c)+{\mu }_{1}n-{\mu }_2{n}^k,& x\in \mathrm{\Omega },t>0,\\ & c_t+u\cdot \mathrm{\nabla }c=\mathrm{\Delta }c-c+{\mu }_3{c}^{\alpha }{w}^{1-\alpha },& x\in \mathrm{\Omega },t>0,\\ & w_t+u\cdot \mathrm{\nabla }w=\mathrm{\Delta }w-w+n,& x\in \mathrm{\Omega },t>0,\\ & u_t+\kappa (u\cdot \mathrm{\nabla }u)=\mathrm{\Delta }u-\mathrm{\nabla }P+n\mathrm{\nabla }\mathrm{\Phi },\mathrm{\nabla }\cdot u=0,& x\in \mathrm{\Omega },t>0,\end{array}$$

in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N(N=2,3)$ with no-flux boundary for $n,c,w$ and no-slip boundary for $u$, where the parameters $\chi >0,\alpha \in (0,1),{\mu }_1\in \mathbb{R},{\mu }_2\ge 0,{\mu }_3>0$ and $\kappa \in \{0,1\},k\ge N$. Due to the interference of the fractional nonlinear term of the Solow-Swan model, we use the Moser-Trudinger inequality to obtain the global existence of the solution for two-dimensional case without logistic source. For three-dimensional case, we control the required estimation with the help of the negative term of logistic source to obtain the boundedness and asymptotic behavior. In the process of estimating the corresponding term, we find the order of the negative term of the logistic source is related to the spatial dimension, and we give the decay estimate of the corresponding solutions when ${\mu }_1<0$ or ${\mu }_1=0,{\mu }_2>0$.