Blow-up prevention by quadratic degradation in a two-dimensional Keller–Segel–Navier–Stokes system

Abstract

This paper deals with an initial-boundary value problem in a two-dimensional smoothly bounded domain for the Keller–Segel–Navier–Stokes system with logistic source, as given by

$$\begin{aligned} \left\{ \begin{array}{rcl} n_t + u\cdot \nabla n &{} =&{} \Delta n - \nabla \cdot (n \nabla c)+rn-\mu n^2,\\ c_t + u\cdot \nabla c &{}=&{} \Delta c -c +n,\\ u_t + u\cdot \nabla u &{}=&{} \Delta u -\nabla P+n \nabla \phi +g,\\ \nabla \cdot u &{} =&{} 0, \end{array} \right. \end{aligned}$$

which describes the mutual interaction of chemotactically moving microorganisms and their surrounding incompressible fluid. It is shown that whenever $\mu >0$, $r\ge 0$, $g\in C^1(\bar{\Omega }\times [0,\infty )) \cap L^\infty (\Omega \times (0,\infty ))$ and the initial data $(n_0, c_0, u_0)$ are sufficiently smooth fulfilling $n_0\not \equiv 0$, the considered problem possesses a global classical solution which is bounded. Moreover, if $r=0$, then this solution satisfies

$$\begin{aligned} n(\cdot ,t)\rightarrow 0 \quad \hbox {and} \quad c(\cdot ,t)\rightarrow 0 \quad \hbox {in } L^\infty (\Omega ) \end{aligned}$$

as $t\rightarrow \infty$, and if additionally $\int \limits _0^\infty \int \limits _\Omega |g(x,t)|^2 \hbox {d}x\hbox {d}t < \infty$, then all solution components decay in the sense that

$$\begin{aligned} n(\cdot ,t)\rightarrow 0, \quad c(\cdot ,t)\rightarrow 0 \quad \hbox {and} \quad u(\cdot ,t)\rightarrow 0 \quad \hbox {in } L^\infty (\Omega ) \end{aligned}$$

as $t\rightarrow \infty$.