Large Time Behavior in a Fractional Chemotaxis-Navier-Stokes System with Competitive Kinetics

Abstract

In this paper, we consider the two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics and fractional diffusion of order $s\in (\frac{1}{2},1)$

$$ \left \{ \textstyle\begin{array}{l} (n_{1})_{t}+u\cdot \nabla n_{1}=-(-\Delta )^{s}n_{1}-\chi _{1}\nabla \cdot (n_{1}\nabla c)+\mu _{1}n_{1}(1-n_{1}-a_{1}n_{2}), \\ (n_{2})_{t}+u\cdot \nabla n_{2}=-(-\Delta )^{s}n_{2}-\chi _{2}\nabla \cdot (n_{2}\nabla c)+\mu _{2}n_{2}(1-a_{2}n_{1}-n_{2}), \\ c_{t}+u\cdot \nabla c=\Delta c-(\alpha n_{1}+\beta n_{2})c, \\ u_{t}+(u\cdot \nabla )u=\Delta u+\nabla P+(\gamma n_{1}+\delta n_{2}) \nabla \phi ,~~~\nabla \cdot u=0 \\ \end{array}\displaystyle \right . $$

on the three dimensional periodic torus $\mathbb{T}^{3}$. Due to the influence of the Navier-Stokes equation, we cannot get the existence of the classical solution to above system in the three-dimensional case. Instead, we investigate the global existence of the weak solution of the system with weaker cell diffusion and arrive at the eventual smoothness of the solution. Furthermore, by constructing different Lyapunov functionals, we arrive at the following asymptotic stability:

• When $a_{1},a_{2}\in (0,1)$, the solution components $(n_{1},n_{2},c,u)\rightarrow (\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}},0,0)$ as $t\rightarrow \infty$;

• When $a_{1}\geq 1>a_{2}>0$, the solution components $(n_{1},n_{2},c,u)\rightarrow (0,1,0,0)$ as $t\rightarrow \infty$;

• When $a_{2}\geq 1>a_{1}>0$, the solution components $(n_{1},n_{2},c,u)\rightarrow (1,0,0,0)$ as $t\rightarrow \infty$.