Small-mass solutions in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with indirect nutrient consumption
Ai Huang, Peter Y. H. Pang, Yifu Wang
TL;DR
The authors study a two-dimensional logarithmic chemotaxis-Navier–Stokes system with indirect nutrient consumption, proving global existence and exponential stabilization of bounded classical solutions under a smallness condition on the initial bacterial mass $m=\int_\Omega n_0$. They introduce a regularized problem and a Lyapunov-type energy functional incorporating entropy $\int_\Omega n\ln\frac{n}{\overline{n_0}}$, the gradient energy of $z=-\ln(v/\|v_0\|_{L^\infty})$, the variance of $w$, and the kinetic energy of $u$, and employ Moser–Trudinger-type inequalities to obtain $\\varepsilon$-independent bounds and dissipation that yield exponential decay. The indirect nutrient consumption mechanism yields higher regularity than direct consumption, with $n(\cdot,t) \to \overline{n_0}$, $v(\cdot,t) \to 0$, $w(\cdot,t) \to \overline{n_0}$, and $u(\cdot,t)$ decaying in $D(A^\beta)$ as $t\to\infty$. The results advance the PDE theory for chemotaxis-fluid systems with tensor-valued singular sensitivities by providing explicit decay rates and a robust analytic framework based on Neumann heat semigroup estimates, Stokes theory, and energy-dissipation methods.
Abstract
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 $Ω\subset\mathbb{R}^2$ under no-flux/Neumann/Neumann/Dirichlet boundary conditions, where $Φ\in W^{2,\infty}(Ω)$, and $S: \overlineΩ\times [0,\infty) \times (0,\infty)\rightarrow\mathbb{R}^{2\times 2}$ is a suitably smooth function that satisfies $|S(x,n,v)|\leq S_0(v) /v $ for all $(x,n,v) \in Ω\times (0,\infty)^2$ with some nondecreasing $S_0: (0,\infty)\rightarrow(0,\infty)$. For all reasonably regular initial data with a smallness assumption merely involving the quantity $\int_Ωn_0$, it is shown that the problem possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state $( \frac{1}{|Ω|}\int_Ωn_0,0,\frac{1}{|Ω|}\int_Ωn_0,0)$ with respect to the norm in $L^\infty(Ω)$. 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.