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: $n_t+u\cdot \nabla n=\hat{I}”n-\nabla \cdot (nS(x,n,v)\cdot \nabla v);v_t+u\cdot \nabla v=\hat{I}”v-vw;w_t+u\cdot \nabla w=\hat{I}”w-w+n;u_t+(u\cdot \nabla )u=\hat{I}”u-\nabla P+n\nabla \hat{I}¦;\nabla \cdot u=0$ in a smooth bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ under no-flux/Neumann/Neumann/Dirichlet boundary conditions, where $\hat{I}¦\in W^{2,\infty }(\hat{I}\text{c◯})$, and $S:\overline{\hat{I}}\text{c◯}\times [0,\infty )\times (0,\infty )\to {\mathbb{R}}^{2\times 2}$ is a suitably smooth function that satisfies $|S(x,n,v)|\le S_0(v)/v$ for all $(x,n,v)\in \hat{I}\text{c◯}\times (0,\infty {)}^2$ with some nondecreasing $S_0:(0,\infty )\to (0,\infty )$. For all reasonably regular initial data with a smallness assumption merely involving the quantity ${\int }_{\hat{I}}\text{c◯}{n}_0$, it is shown that the problem possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state $(\frac{1}{|\hat{I}\text{c◯}|}{\int }_{\hat{I}}\text{c◯}{n}_0,0,\frac{1}{|\hat{I}\text{c◯}|}{\int }_{\hat{I}}\text{c◯}{n}_0,0)$ with respect to the norm in $L^{\infty }(\hat{I}\text{c◯})$. 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.

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