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

Abstract

The chemotaxis–Navier–Stokes system

is considered under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$, in a bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with smooth boundary, where $\mathrm{\Phi }\in W^{2,\infty }(\mathrm{\Omega })$, and where $f\in C^1([0,\infty ))$ and $\chi \in C^2([0,\infty ))$ are nonnegative with $f(0)=0$. 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 ($\star$), 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 $f$ and $\chi$, inter alia allowing for the prototypical case when

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 $L^2$ dissipation property of the fluid evolution with a quasi-dissipative structure associated with the chemotaxis subsystem in ($\star$).