Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth
Authors: Braukhoff Marcel 
Published: September 2016

In biology, the behaviour of a bacterial suspension in an incompressible fluid drop is modelled by the chemotaxis-Navier–Stokes equations. This paper introduces an exchange of oxygen between the drop and its environment and an additionally logistic growth of the bacteria population. A prototype system is given by

\left\{\begin{align}
n_{t} + u \cdot \mathrm{∇}n & = \mathrm{\Delta }n−\mathrm{∇} \cdot (n\mathrm{∇}c) + n−n^{2}, && x \in \mathrm{\Omega },\:t > 0, \\
c_{t} + u \cdot \mathrm{∇}c & = \mathrm{\Delta }c−nc, && x \in \mathrm{\Omega },\:t > 0, \\
u_{t} & = \mathrm{\Delta }u + u \cdot \mathrm{∇}u + \mathrm{∇}P−n\mathrm{∇}\varphi , && x \in \mathrm{\Omega },\:t > 0, \\
\mathrm{∇} \cdot u & = 0, && x \in \mathrm{\Omega },\:t > 0 \\ \end{align}\right.

in conjunction with the initial data

(n,c,u)( \cdot ,0) = (n_{0},c_{0},u_{0})

and the boundary conditions

\begin{matrix} \frac{\partial c}{\partial \nu } & = 1−c,\:\frac{\partial n}{\partial \nu } = n\frac{\partial c}{\partial \nu },\:u = 0, & \:x \in \partial \mathrm{\Omega },\:t > 0. \\ \end{matrix}

Here, the fluid drop is described by

\mathrm{\Omega } \subset \mathbb{R}^{N}

being a bounded convex domain with smooth boundary. Moreover,

φ

is a given smooth gravitational potential.

Requiring sufficiently smooth initial data, the existence of a unique global classical solution for

N = 2

is proved, where

\left\|n\right\|_{L^{p}(\mathrm{\Omega })}

is bounded in time for all

p < \infty

, as well as the existence of a global weak solution for

N = 3

.

Le comportement d'une suspension bactérienne dans une goutte de liquide incompressible est décrit par les équations de chemotaxis-Navier–Stokes. Cet article introduit un échange d'oxygène entre la goutte et son environnement et une croissance logistique de la population bactérienne. Le système généralise le prototype

\left\{\begin{align}
n_{t} + u \cdot \mathrm{∇}n & = \mathrm{\Delta }n−\mathrm{∇} \cdot (n\mathrm{∇}c) + n−n^{2}, && x \in \mathrm{\Omega },\:t > 0, \\
c_{t} + u \cdot \mathrm{∇}c & = \mathrm{\Delta }c−nc, && x \in \mathrm{\Omega },\:t > 0, \\
u_{t} & = \mathrm{\Delta }u + u \cdot \mathrm{∇}u + \mathrm{∇}P−n\mathrm{∇}\varphi , && x \in \mathrm{\Omega },\:t > 0, \\
\mathrm{∇} \cdot u & = 0, && x \in \mathrm{\Omega },\:t > 0 \\ \end{align}\right.

associé à la donnée initiale

(n,c,u)( \cdot ,0) = (n_{0},c_{0},u_{0})

et aux conditions du bord

\begin{matrix} \frac{\partial c}{\partial \nu } & = 1−c,\:\frac{\partial n}{\partial \nu } = n\frac{\partial c}{\partial \nu },\:u = 0, & \:x \in \partial \mathrm{\Omega },\:t > 0 \\ \end{matrix}

d'où

\mathrm{\Omega } \subset \mathbb{R}^{N}

soit un domaine borné et convexe avec un bord lisse. En outre,

φ

soit un potentiel lisse gravitationnel. En supposant que la donnée initiale soit suffisamment régulière, on démontre l'existence d'une solution classique unique pour

N = 2

telle que

\left\|n\right\|_{L^{p}(\mathrm{\Omega })}

est borné pour

p < \infty

et l'existence d'une solution faible globale pour

N = 3

.