How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?

Abstract:

The chemotaxis-Navier-Stokes system \begin{equation*} (\star )\qquad \qquad \qquad \quad \begin {cases} n_t + u\cdot \nabla n & =\ \ \Delta n - \nabla \cdot (n\chi (c)\nabla c),\\[1mm] c_t + u\cdot \nabla c & =\ \ \Delta c-nf(c), \\[1mm] u_t + (u\cdot \nabla )u & =\ \ \Delta u + \nabla P + n \nabla \Phi , \\[1mm] \nabla \cdot u & =\ \ 0 \end{cases} \qquad \qquad \qquad \quad \end{equation*} is considered under boundary conditions of homogeneous Neumann type for $n$ and $c$, and Dirichlet type for $u$, in a bounded convex domain $\Omega \subset \mathbb {R}^3$ with smooth boundary, where $\Phi \in W^{1,\infty }(\Omega )$ and $\chi$ and $f$ are sufficiently smooth given functions generalizing the prototypes $\chi \equiv const.$ and $f(s)=s$ for $s\ge 0$.

It is known that for all suitably regular initial data $n_0, c_0$ and $u_0$ satisfying $0\not \equiv n_0\ge 0$, $c_0\ge 0$ and $\nabla \cdot u_0=0$, a corresponding initial-boundary value problem admits at least one global weak solution which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. The present paper shows that after some relaxation time, this solution enjoys further regularity properties and thereby complies with the concept of eventual energy solutions, which is newly introduced here and which inter alia requires that two quasi-dissipative inequalities are ultimately satisfied.

Moreover, it is shown that actually for any such eventual energy solution $(n,c,u)$ there exists a waiting time $T_0\in (0,\infty )$ with the property that $(n,c,u)$ is smooth in $\bar \Omega \times [T_0,\infty )$ and that \begin{eqnarray*} n(x,t)\to \overline {n_0}, \qquad c(x,t)\to 0 \qquad \mbox {and} \qquad u(x,t)\to 0 \end{eqnarray*} hold as $t\to \infty$, uniformly with respect to $x\in \Omega$. This resembles a classical result on the three-dimensional Navier-Stokes system, asserting eventual smoothness of arbitrary weak solutions thereof which additionally fulfill the associated natural energy inequality. In consequence, our results inter alia indicate that under the considered boundary conditions, the possibly destabilizing action of chemotactic cross-diffusion in ($\star$) does not substantially affect the regularity properties of the fluid flow at least on large time scales.