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$\{\begin{array}{cccc}{n}_t+u\cdot \nabla n& =\nabla \cdot \left(D\right(n)\nabla n-nS(x,n,c)\cdot \nabla c),& x\in \Omega ,& t>0,\\ c_t+u\cdot \nabla c& =\Delta c-cn,& x\in \Omega ,& t>0,\\ u_t+(u\cdot \nabla )u& =\Delta u+\nabla P+n\nabla \Phi ,\nabla \cdot u=0,& x\in \Omega ,& t>0,\\ \left(D\right(n)\nabla n-n& S(x,n,c)\cdot \nabla c)\cdot \nu =\nabla c\cdot \nu =0,u=0,& x\in \partial \Omega ,& t>0,\\ n(\cdot ,0)=n_0,& c(\cdot ,0)=c_0,u(\cdot ,0)=u_0,& x\in \Omega ,\end{array}$ in a smoothly bounded domain $\Omega \subset R^2$. Assuming $S:\bar{\Omega }\times [0,\infty )\times (0,\infty )\to R^{2\times 2}$ to be sufficiently regular and such that with $\gamma \in [0,\frac{5}{6}]$ and some non-decreasing $S_0:(0,\infty )\to (0,\infty )$, we have$\left|S\right(x,n,c\left)\right|\le \frac{S_0\left(c\right)}{c^{\gamma }}\text{for all }(x,n,c)\in \bar{\Omega }\times [0,\infty )\times (0,\infty ),$ 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 $L\left(M\right)>0$ such that whenever$\underset{n\to \infty }{\lim \inf }D\left(n\right)>L\text{and}\underset{n\searrow 0}{\lim \inf }\frac{D\left(n\right)}{n}>0$ are satisfied and the initial data $(n_0,c_0,u_0)$ are suitably regular and satisfy ${\Vert c_0\Vert }_{L^{\infty }\left(\Omega \right)}\le M$ there is a global and bounded weak solution for the initial-boundary value problem above. Under the additional assumption of $D\left(0\right)>0$ this solution is moreover a classical solution of the same problem.