JournalsaihpcVol. 33, No. 5pp. 1329–1352

Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system

  • Michael Winkler

    Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany
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Abstract

The chemotaxis–Navier–Stokes system

{nt+un=Δn(nχ(c)c),ct+uc=Δcnf(c),ut+(u)u=Δu+P+nΦ,u=0,()\left\{\begin{matrix} n_{t} + u \cdot \mathrm{∇}n\: & = \: & \mathrm{\Delta }n−\mathrm{∇} \cdot (n\chi (c)\mathrm{∇}c),\: \\ c_{t} + u \cdot \mathrm{∇}c\: & = \: & \mathrm{\Delta }c−nf(c),\: \\ u_{t} + (u \cdot \mathrm{∇})u\: & = \: & \mathrm{\Delta }u + \mathrm{∇}P + n\mathrm{∇}\mathrm{\Phi },\: \\ \mathrm{∇} \cdot u\: & = \: & 0,\: \\ \end{matrix}\right.\:\:( \star )

is considered under homogeneous boundary conditions of Neumann type for nn and cc, and of Dirichlet type for uu, in a bounded convex domain ΩR3\mathrm{\Omega } \subset \mathbb{R}^{3} with smooth boundary, where ΦW2,(Ω)\mathrm{\Phi } \in W^{2,\infty }(\mathrm{\Omega }), and where fC1([0,)f \in C^{1}([0,\infty) and χC2([0,)\chi \in C^{2}([0,\infty) are nonnegative with f(0)=0f(0) = 0. Problems of this type have been used to describe the mutual interaction of populations of swimming aerobic bacteria with the surrounding fluid. Up to now, however, global existence results seem to be available only for certain simplified variants such as e.g. the two-dimensional analogue of (⋆), or the associated chemotaxis–Stokes system obtained on neglecting the nonlinear convective term in the fluid equation.

The present work gives an affirmative answer to the question of global solvability for (\star) in the following sense: Under mild assumptions on the initial data, and under modest structural assumptions on ff and χ\chi, inter alia allowing for the prototypical case when

f(s)=sfor alls0andχconst.,f(s) = s\:\text{for all}s \geq 0\:\text{and}\:\chi \equiv \mathrm{const}.,

the corresponding initial–boundary value problem is shown to possess a globally defined weak solution.

This solution is obtained as the limit of smooth solutions to suitably regularized problems, where appropriate compactness properties are derived on the basis of a priori estimates gained from an energy-type inequality for (\star) which in an apparently novel manner combines the standard L2L^{2} dissipation property of the fluid evolution with a quasi-dissipative structure associated with the chemotaxis subsystem in (\star).

Cite this article

Michael Winkler, Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, pp. 1329–1352

DOI 10.1016/J.ANIHPC.2015.05.002