Global Solutions to the Coupled Chemotaxis-Fluid Equations

Duan, Renjun; Lorz, Alexander; Markowich, Peter A.

doi:10.1080/03605302.2010.497199

Cited by 358 publications

(208 citation statements)

References 25 publications

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“…The only global existence results we are aware of concentrate on the construction of solutions near constant steady states ( [8]), or on the particular case when χ precisely coincides with a multiple of f ( [3]), where the latter not only excludes the situation determined by (1.4), but under the natural assumption that f (0) = 0 apparently also rules out any choice of χ which is consistent with standard approaches in the modeling of chemotaxis phenomena ( [13]). …”

Section: Resultsmentioning

confidence: 99%

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

Winkler

2013

Arch Rational Mech Anal

385

162

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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 (⋆).

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Section: Resultsmentioning

confidence: 99%

“…The novelty of the present reasoning, as compared to previous approaches based on this or similar identities (cf. also [8]), appears to consist in the particular manner in which (3.1) will subsequently be related to the natural dissipative properties of the Navier-Stokes subsystem in (2.9).…”

Section: (215)mentioning

confidence: 99%

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

Winkler

2013

Arch Rational Mech Anal

385

162

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“…Noting that 4 3 ≤ 6−6α 2+3α < 2 (which is obtained from the condition 1/6 < α ≤ 1/3), we obtain, from Young's inequality and Gagliardo-Nierenberg inequality, the following :…”

Section: Weak Solutionsmentioning

confidence: 99%

“…[1], [2], [4], [9] and [15] for the case that α = 0 and references therein). It was proved in [5] that weak solutions of (KSS) for bounded domains exist in case that α ∈ (1/2, 1] in two dimensions or in case that α ∈ ( −5+ √…”

Section: Introductionmentioning

confidence: 99%

Existence of global solutions for a chemotaxis-fluid system with nonlinear diffusion

Chung

Kang

2016

Journal of Mathematical Physics

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We consider a coupled system consisting of the Navier-Stokes equations and a porous medium type of Keller-Segel system that model the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global-in-time existence of weak solutions for the Cauchy problem of the system in dimension three. In addition, if the Stokes system, instead Navier-Stokes system, is considered for the fluid equation, we prove that bounded weak solutions exist globally in time. AMS Subject Classification: 35Q30, 35Q35

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“…More recently, the interaction between the Cucker-Smale type flocking particle and the compressible viscous fluid is studied in [4]. For the other related particle interacting with fluid, we refer to [6,7,8,9,13,17,22,26].…”

Section: Introductionmentioning

confidence: 99%

Two-species flocking particles immersed in a fluid

Choi

Kwon

2013

Communications in Information and Systems

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We present a new particle-fluid model describing the motion of two-species flocking particles immersed in an incompressible viscous fluid. The flocking particles are directly affected by the incompressible fluid through a drag force, and they are also coupled with each other via the fluid. On the other hand, the two-species particles are coupled with each other via the viscous fluid. For this proposed model, we show the global existence of a unique strong solution when the initial data is sufficiently small, and we also investigate the large-time behavior of the solutions under suitable conditions.

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Global Solutions to the Coupled Chemotaxis-Fluid Equations

Cited by 358 publications

References 25 publications

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

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

Existence of global solutions for a chemotaxis-fluid system with nonlinear diffusion

Two-species flocking particles immersed in a fluid

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