Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system

We study the chemotaxis-fluid system \begin{align*} \left{ \begin{array}{r@{,}c@{,}c@{\ }l@{\quad}l@{\quad}l@{,}c} n_{t}&+&u\cdot!\nabla n&=\Delta n-\nabla!\cdot(\frac{n}{c}\nabla c),
&x\in\Omega,& t>0, c_{t}&+&u\cdot!\nabla c&=\Delta c-nc,\ &x\in\Omega,& t>0, u_{t}&+&\nabla P&=\Delta u+n\nabla\phi,\ &x\in\Omega,& t>0, &&\nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*} under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$, where $\mathrm{\Omega }\subset {\mathbb{R}}^2$ is a bounded domain with smooth boundary and $\varphi \in C^2\left(\bar{\mathrm{\Omega }}\right)$. From recent results it is known that for suitable regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution. We will show that for small initial mass ${\int }_{\mathrm{\Omega }}\text{\hspace{0.17em}⁣}{n}_0$ these generalized solutions will eventually become classical solutions of the system and obey certain asymptotic properties. Moreover, from the analysis of certain energy-type inequalities arising during the investigation of the eventual regularity, we will also derive a result on global existence of classical solutions under assumption of certain smallness conditions on the size of $n_0$ in $L^1\text{\hspace{0.17em}⁣}\left(\mathrm{\Omega }\right)$ and in $L\log L\text{\hspace{0.17em}⁣}\left(\mathrm{\Omega }\right)$, $u_0$ in $L^4\text{\hspace{0.17em}⁣}\left(\mathrm{\Omega }\right)$, and of $\mathrm{\nabla }{c}_0$ in $L^2\text{\hspace{0.17em}⁣}\left(\mathrm{\Omega }\right)$.