Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

Abstract

The chemotaxis-Navier-Stokes system          n t + u · ∇n = ∆n − ∇ · (nχ(c)∇c), c t + u · ∇c = ∆c − nf (c),is considered under homogeneous boundary conditions of Neumann type for n and c, and of Dirichlet type for u, in a bounded convex domain Ω ⊂ R 3 with smooth boundary, where Φ ∈ W 1,∞ (Ω), and where f ∈ C 1 ([0, ∞)) and χ ∈ C 2 ([0, ∞)) 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 (⋆), 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 (⋆) in the following sense: Under mild assumptions on the initial data, and under modest structural assumptions on f and χ, inter alia allowing for the prototypical case when f (s) = s for all s ≥ 0 and χ ≡ 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 (⋆) 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 (⋆).

Long-term behaviour in a chemotaxis-fluid system with logistic source

“…Fortunately, we already have established that c ε (t) L ∞ (Ω) converges to 0. The same quantity has proven useful in the derivation of estimates for n(t) L p (Ω) for large t already in [45,Sec. 5] and [42,Sec.…”

“…Global weak solutions on Ω = R 2 have been treated in [50] under weaker conditions on the initial data. In the setting of [44], the convergence of solutions to the stationary state was proven in [45]; its rate was given in [48]. Upon neglection of the nonlinear fluid term (u · ∇)u, that is upon consideration of Stokes flow instead of a Navier-Stokes governed fluid, global weak solutions can also be found in bounded three-dimensional domains ( [44]).…”

“…Upon neglection of the nonlinear fluid term (u · ∇)u, that is upon consideration of Stokes flow instead of a Navier-Stokes governed fluid, global weak solutions can also be found in bounded three-dimensional domains ( [44]). (The results of [44,45] have been extended to non-convex domains in [18].) For the three-dimensional setting (of bounded convex domains) with full Navier-Stokes-fluid and large initial data only recently the existence of weak solutions has been demonstrated by Winkler ([41]).…”

Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system

“…[4,28,30,31], for instance). Since due to the dampening effect of signal absorption, the structure of these systems is somewhat different from that of (1.1) with signal production, the mathematical approaches already developed for these chemotaxis-fluid systems can apparently not be applied to (1.1).…”

Singular sensitivity in a Keller–Segel-fluid system

“…A prototypical and popular choice of the functions in the above system is accordingly given by g(n, c) = −nc and the classical χ(n, c) = χ = const. This choice was, for convex domains, first covered by the existence results in [39], also extension to nonconvex domains ( [16]), results on convergence of solutions ( [40]) and its rate ( [46]) are available. The most advanced developments dealing with the full chemotaxis-Navier-Stokes system in three-dimensional domains are constituted by the construction of global weak solutions in [43] and the proof of their eventual regularization and convergence in [44].…”