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Transactions of the American Mathematical Society

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

Author: Michael Winkler
Journal: Trans. Amer. Math. Soc. 369 (2017), 3067-3125
MSC (2010): Primary 35B65, 35B40; Secondary 35K55, 92C17, 35Q30, 35Q92
Published electronically: July 29, 2016
MathSciNet review: 3605965
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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.

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Additional Information

Michael Winkler
Affiliation: Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany
MR Author ID: 680863

Keywords: Chemotaxis, Navier-Stokes, global existence, boundedness, eventual regularity, stabilization, entropy dissipation
Received by editor(s): February 16, 2015
Received by editor(s) in revised form: April 21, 2015
Published electronically: July 29, 2016
Article copyright: © Copyright 2016 American Mathematical Society