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

Abstract

This paper is concerned with the singular chemotaxis-fluid system with indirect nutrient consumption: $ n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(n S(x,n,v)\cdot \nabla v);\ v_{t}+u\cdot\nabla v=\Delta v-vw;\ w_{t}+u\cdot\nabla w=\Delta w-w+n;\ u_t+(u\cdot\nabla) u=\Delta u-\nabla P+n\nabla\Phi;\ \nabla\cdot u=0\ $ in a smooth bounded domain $\Omega\subset\mathbb{R}^2$ under no-flux/Neumann/Neumann/Dirichlet boundary conditions, where $\Phi\in W^{2,\infty}(\Omega)$, and $S: \overline{\Omega}\times [0,\infty) \times (0,\infty)\rightarrow\mathbb{R}^{2\times 2}$ is a suitably smooth function that satisfies $|S(x,n,v)|\leq S_0(v) /v $ for all $(x,n,v) \in \Omega\times (0,\infty)^2$ with some nondecreasing $S_0: (0,\infty)\rightarrow(0,\infty)$. For all reasonably regular initial data with a smallness assumption merely involving the quantity $\int_\Omega n_0$, it is shown that the problem possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state $( \frac{1}{|\Omega|}\int_{\Omega}n_0,0,\frac{1}{|\Omega|}\int_{\Omega}n_0,0)$ with respect to the norm in $L^\infty(\Omega)$. 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.