Small-mass solutions in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with indirect nutrient consumption

This paper is concerned with the singular chemotaxis-fluid system with indirect nutrient consumption: n_{t}+u\cdot\nabla n=Δn-\nabla\cdot(n S(x,n,v)\cdot \nabla v);\ v_{t}+u\cdot\nabla v=Δv-vw;\ w_{t}+u\cdot\nabla w=Δw-w+n;\ u_t+(u\cdot\nabla) u=Δu-\nabla P+n\nablaΦ;\ \nabla\cdot u=0\ in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ under no-flux/Neumann/Neumann/Dirichlet boundary conditions, where $\mathrm{\Phi }\in W^{2,\mathrm{\infty }}(\mathrm{\Omega })$, and $S:\bar{\mathrm{\Omega }}\times [0,\mathrm{\infty })\times (0,\mathrm{\infty })\to {\mathbb{R}}^{2\times 2}$ is a suitably smooth function that satisfies $\mathrm{\mid }S(x,n,v)\mathrm{\mid }\le S_0(v)\mathrm{/}v$ for all $(x,n,v)\in \mathrm{\Omega }\times (0,\mathrm{\infty }{)}^2$ with some nondecreasing $S_0:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$. For all reasonably regular initial data with a smallness assumption merely involving the quantity ${\int }_{\mathrm{\Omega }}{n}_0$, it is shown that the problem possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state $(\frac{1}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\int }_{\mathrm{\Omega }}{n}_0,0,\frac{1}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\int }_{\mathrm{\Omega }}{n}_0,0)$ with respect to the norm in $L^{\mathrm{\infty }}(\mathrm{\Omega })$. This rigorously confirms that, at least in the two-dimensional setting, in comparison to the direct mechanism of nutrient consumption, an indirect mechanism can induce much more regularity of solutions to the chemotaxis–fluid system even with a singular tensor-valued sensitivity.