Asymptotic profile of a two-dimensional chemotaxis--Navier--Stokes system with singular sensitivity and logistic source

The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1} \left\{\begin{array}{ll} n_t+u\cdot \nabla n=\triangle n-\chi\nabla\cdotp \left(\displaystyle\frac n {c}\nabla c\right)+n(r-\mu n), c_t+u\cdot \nabla c=\triangle c-nc, u_t+ (u\cdot \nabla) u=\Delta u+\nabla P+n\nabla\phi, \nabla\cdot u=0, \end{array}\right. \end{equation*} is considered in a bounded smooth domain $\Omega \subset \mathbb{R}^2$, where $\phi\in W^{1,\infty}(\Omega)$, $\chi>0$, $r\in \mathbb{R}$ and $\mu> 0$ are given parameters. It is shown that there exists a value $\mu_*(\Omega,\chi, r)\geq 0$ such that whenever $\mu>\mu_*(\Omega,\chi, r)$, the global-in-time classical solution to the system is uniformly bounded with respect to $x\in \Omega$. Moreover, for the case $r>0$, $(n,c,\frac {|\nabla c|}c,u)$ converges to $(\frac r \mu,0,0,0)$ in $L^\infty(\Omega)\times L^\infty(\Omega)\times L^p(\Omega)\times L^\infty(\Omega)$ for any $p>1$ exponentially as $t\rightarrow \infty$, while in the case $r=0$, $(n,c,\frac {|\nabla c|}c,u)$ converges to $(0,0,0,0)$ in $(L^\infty(\Omega))^4$ algebraically. To the best of our knowledge, these results provide the first precise information on the asymptotic profile of solutions in two dimensions.