Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

Abstract

We consider the chemotaxis-fluid system

(0.1)

in a bounded convex domain $\Omega \subset {\mathbb {R}}^3$ with smooth boundary, where $\phi \in W^{1,\infty }(\Omega )$ and D, f and S are given functions with values in $[0,\infty ), [0,\infty )$ and ${\mathbb {R}}^{3\times 3}$, respectively. In the existing literature, the derivation of results on global existence and qualitative behavior essentially relies on the use of energy-type functionals which seem to be available only in special situations, necessarily requiring the matrix-valued S to actually reduce to a scalar function of c which, along with f, in addition should satisfy certain quite restrictive structural conditions. The present work presents a novel a priori estimation method which allows for removing any such additional hypothesis: besides appropriate smoothness assumptions, in this paper it is only required that f is locally bounded in $[0,\infty )$, that S is bounded in $\Omega \times [0,\infty )^2$, and that $D(n)\ge k_{D}n^{m-1}$ for all $n\ge 0$ with some $k_{D}>0$ and some

$$\begin{aligned} m>\frac{7}{6}. \end{aligned}$$

It is shown that then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a globally defined weak solution. The method introduced here is efficient enough to moreover provide global boundedness of all solutions thereby obtained in that, inter alia, $n\in L^\infty (\Omega \times (0,\infty ))$. Building on this boundedness property, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium $(\overline{n_0},0,0)$ in an appropriate sense, where $\overline{n_0}:=\frac{1}{|\Omega |} \int _{\Omega }n_0$, provided that merely $n_0\not \equiv 0$ and $f>0$ on $(0,\infty )$. To the best of our knowledge, these are the first results on boundedness and asymptotics of large-data solutions in a three-dimensional chemotaxis-fluid system of type (0.1).