Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth

Braukhoff, Marcel

doi:10.1016/j.anihpc.2016.08.003

Braukhoff, Marcel

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 1013-1039.

Résumé
Abstract

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

{\begin{matrix} n_{t} + u \cdot \nabla n & = Δ n - \nabla \cdot (n \nabla c) + n - n^{2}, & x \in Ω, t > 0, \\ c_{t} + u \cdot \nabla c & = Δ c - n c, & x \in Ω, t > 0, \\ u_{t} & = Δ u + u \cdot \nabla u + \nabla P - n \nabla φ, & x \in Ω, t > 0, \\ \nabla \cdot u & = 0, & x \in Ω, t > 0 \end{matrix}

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 ν} & = 1 - c, \frac{\partial n}{\partial ν} = n \frac{\partial c}{\partial ν}, u = 0, & x \in \partial Ω, t > 0 \end{matrix}

d'où

Ω \subset 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

{‖ n ‖}_{L^{p} (Ω)}

est borné pour

p < \infty

et l'existence d'une solution faible globale pour

N = 3

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

{\begin{matrix} n_{t} + u \cdot \nabla n & = Δ n - \nabla \cdot (n \nabla c) + n - n^{2}, & x \in Ω, t > 0, \\ c_{t} + u \cdot \nabla c & = Δ c - n c, & x \in Ω, t > 0, \\ u_{t} & = Δ u + u \cdot \nabla u + \nabla P - n \nabla φ, & x \in Ω, t > 0, \\ \nabla \cdot u & = 0, & x \in Ω, t > 0 \end{matrix}

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 ν} & = 1 - c, \frac{\partial n}{\partial ν} = n \frac{\partial c}{\partial ν}, u = 0, & x \in \partial Ω, t > 0 . \end{matrix}

Here, the fluid drop is described by

Ω \subset 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 ${‖ n ‖}_{L^{p} (Ω)}$ is bounded in time for all $p < \infty$ , as well as the existence of a global weak solution for $N = 3$ .

MR Zbl

DOI : 10.1016/j.anihpc.2016.08.003

Mots clés : Chemotaxis, Navier–Stokes, Non-homogeneous boundary conditions, Logistic growth

@article{AIHPC_2017__34_4_1013_0,
     author = {Braukhoff, Marcel},
     title = {Global (weak) solution of the {chemotaxis-Navier{\textendash}Stokes} equations with non-homogeneous boundary conditions and logistic growth},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1013--1039},
     publisher = {Elsevier},
     volume = {34},
     number = {4},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.08.003},
     mrnumber = {3661869},
     zbl = {1417.92028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.08.003/}
}

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Braukhoff, Marcel. Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 1013-1039. doi : 10.1016/j.anihpc.2016.08.003. http://www.numdam.org/articles/10.1016/j.anihpc.2016.08.003/

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