Abstract

We study the following chemotaxis-Navier–Stokes system with general sensitivity and nonlinear production ⋆ $$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nf(n)\nabla c), \\&c_t+u\cdot \nabla c=\Delta c-c+g(n),\\&u_t+(u\cdot \nabla )u+\nabla P=\Delta u+n \nabla \phi , ~ \nabla \cdot u=0 \end{aligned} \right. \end{aligned}$$ in a bounded domain \(\Omega \subset \mathbb {R}^2\) , where the chemotaxis sensitivity function \(f\in C^2([0,\infty ))\) satisfies that \(|f(s)|\le K_f(1+s)^{-\alpha }\) for all \(s\ge 0\) with \(K_f>0\) and \(\alpha \in \mathbb {R}\) , and the signal production function \(g\in C^1([0,\infty ))\) is such that \(0\le g(s)\le K_g s(1+s)^{\beta -1}\) for all \(s\ge 0\) with \(K_g,\beta >0\) . It is shown in this paper that for all reasonably regular initial data, the corresponding initial-boundary value problem of ( \(\star \) ) possesses a unique globally bounded classical solution if \(\alpha >\frac{1}{2}[(2\beta -1)_+-1]\) for \(0<\beta <1\) , or if \(\alpha >\beta -1\) for \(\beta \ge 1\) . Our work is one of the few explorations involving chemotaxis-fluid models with nonlinear production mechanisms and greatly extends the global solvability result obtained in Black (Nonlinear Anal Real World Appl 31:593–609, 2016) only for the chemotaxis-Stokes variant of ( \(\star \) ) with \(\alpha =0\) and the sublinear signal production of \(\beta \in (0,1)\) .

中文翻译：

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具有一般灵敏度和非线性产生的二维趋化-纳维-斯托克斯系统中的全局有界解

摘要

我们研究了以下具有一般灵敏度和非线性产生的趋化性-纳维-斯托克斯系统 ⋆ $$\begin{对齐} \left\{ \begin{对齐}&n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nf(n)\nabla c), \\&c_t+u\cdot \nabla c=\Delta c-c+g(n),\\&u_t+(u\cdot \nabla )u+\nabla P=\Delta u+n \nabla \phi , ~ \nabla \cdot u=0 \end {对齐} \右。 \end{aligned}$$ 在有界域\(\Omega \subset \mathbb {R}^2\)中，其中趋化敏感性函数\(f\in C^2([0,\infty ))\)对于所有\(s\ge 0\)且\(K_f>0\)和\(\alpha \ )满足\(|f(s)|\le K_f(1+s)^{-\alpha }\)在 \mathbb {R}\) 中，信号产生函数\(g\in C^1([0,\infty ))\)使得\(0\le g(s)\le K_g s(1 +s)^{\beta -1}\)对于所有\(s\ge 0\)且\(K_g,\beta >0\)。本文表明，对于所有相当规则的初始数据，相应的 ( \(\star \) )初始边值问题拥有唯一的全局有界经典解，如果\(\alpha >\frac{1}{2 }[(2\beta -1)_+-1]\)对于\(0<\beta <1\)，或者如果\(\alpha >\beta -1\)对于\(\beta \ge 1\ ）。我们的工作是涉及具有非线性产生机制的趋化性流体模型的少数探索之一，并且极大地扩展了 Black (Nonlinear Anal Real World Appl 31:593–609, 2016) 中获得的全局可解性结果，仅适用于 ( \(\star \) ) 与\(\alpha =0\)以及\(\beta \in (0,1)\)的次线性信号产生。