Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivity

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This paper considers the following chemotaxis-Stokes system: $$ \left \{ \textstyle\begin{array}{l} n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(\frac{n}{c}\nabla c), c_{t}+u\cdot\nabla c=\Delta c-nc, u_{t}=\Delta u+\nabla P+n\nabla\phi, \nabla\cdot u=0, \end{array}\displaystyle \right . $$ in two-dimensional smoothly bounded domains, which can be seen as a model to describe the migration of aerobic bacteria swimming in an incompressible fluid. It is proved that the corresponding initial-boundary value problem possesses a global generalized solution for any sufficiently regular initial data $(n_{0}, c_{0}, u_{0})$ satisfying $n_{0}\geq0$ and $c_{0}>0$ . Moreover, the solution component c satisfies $c(\cdot,t)\overset{\star}{\rightharpoonup}0$ in $L^{\infty}(\Omega )$ as $t\rightarrow\infty$ and $c(\cdot,t)\rightarrow0$ in $L^{p}(\Omega)$ as $t\rightarrow\infty$ for any $p\in[1,\infty)$ . To the best of our knowledge, this is the first result on global solvability in a chemotaxis-Stokes system with singular sensitivity and signal absorption.