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 $\mathrm{\Omega }\subset {\mathbb{R}}^2$, where $\varphi \in W^{1,\mathrm{\infty }}(\mathrm{\Omega })$, $\chi >0$, $r\in \mathbb{R}$ and $\mu >0$ are given parameters. It is shown that there exists a value ${\mu }_{\ast }(\mathrm{\Omega },\chi ,r)\ge 0$ such that whenever $\mu >{\mu }_{\ast }(\mathrm{\Omega },\chi ,r)$, the global-in-time classical solution to the system is uniformly bounded with respect to $x\in \mathrm{\Omega }$. Moreover, for the case $r>0$, $(n,c,\frac{\mathrm{\mid }\mathrm{\nabla }c\mathrm{\mid }}{c},u)$ converges to $(\frac{r}{\mu },0,0,0)$ in $L^\infty(\Omega)\times L^\infty(\Omega)\times L^p(\Omega)\times L^\infty(\Omega)$forany$p>1$exponentiallyas$t\rightarrow \infty$, while in the case $r=0$, $(n,c,\frac{\mathrm{\mid }\mathrm{\nabla }c\mathrm{\mid }}{c},u)$ converges to $(0,0,0,0)$ in $(L^{\mathrm{\infty }}(\mathrm{\Omega }){)}^4$ algebraically. To the best of our knowledge, these results provide the first precise information on the asymptotic profile of solutions in two dimensions.