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

Marcel Braukhoff

doi:10.1016/j.anihpc.2016.08.003

JournalsaihpcVol. 34, No. 4pp. 1013–1039

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

Marcel Braukhoff
University of Paderborn, Warburgerstraße 100, 33098 Paderborn, Germany

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Abstract

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

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

in conjunction with the initial data $(n, c, u) (\cdot, 0) = (n_{0}, c_{0}, u_{0})$ and the boundary conditions

\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.

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 $N = 3$ .

Résumé

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

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

associé à la donnée initiale $(n, c, u) (\cdot, 0) = (n_{0}, c_{0}, u_{0})$ et aux conditions du bord

\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

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 $N = 3$ .

Cite this article

Marcel Braukhoff, Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 4, pp. 1013–1039

DOI 10.1016/J.ANIHPC.2016.08.003