Very mild diffusion enhancement and singular sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier--Stokes system

We consider an initial-boundary value problem for the chemotaxis-Navier--Stokes system \begin{align*} \left\{ \begin{array}{c@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\nabla n=\nabla\cdot\big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c\big),\ &x\in\Omega,& t>0,\\ c_{t}+u\cdot\nabla c=\Delta c-cn,\ &x\in\Omega,& t>0,\\ u_{t}+(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\Phi,\quad \nabla\cdot u=0,\ &x\in\Omega,& t>0,\\ \big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c)\cdot\nu=\nabla c\cdot\nu=0,\ u=0,\ &x\in\partial\Omega,& t>0,\\ n(\cdot,0)=n_0,\ c(\cdot,0)=c_0,\ u(\cdot,0)=u_0,\ &x\in\Omega. \end{array}\right. \end{align*} in a smoothly bounded domain $\Omega\subset\mathbb{R}^2$. Assuming $S:\overline{\Omega}\times[0,\infty)\times(0,\infty)\rightarrow \mathbb{R}^{2\times 2}$ to be sufficiently regular and such that with $\gamma\in[0,\frac56]$ and some non-decreasing $S_0:(0,\infty)\to(0,\infty)$, we have \begin{align*} \big|S(x,n,c)\big|\leq \frac{S_0(c)}{c^\gamma}\quad\text{for all }(x,n,c)\in\overline{\Omega}\times[0,\infty)\times(0,\infty), \end{align*} we show that if $D:[0,\infty)\to[0,\infty)$ is suitably regular and positive throughout $(0,\infty)$, then for all $M>0$ one can find $L(M)>0$ such that whenever $$\liminf_{n\to\infty} D(n)>L\quad\text{and}\quad \liminf_{n\searrow0}\frac{D(n)}{n}>0$$ are satisfied and the initial data $(n_0,c_0,u_0)$ are suitably regular and satisfy $\|c_0\|_{L^{\infty}(\Omega)}\leq M$ there is a global and bounded weak solution for the initial-boundary value problem above. Under the additional assumption of $D(0)>0$ this solution is moreover a classical solution of the same problem.