Existence and uniqueness theorem on mild solutions to the Keller–Segel system coupled with the Navier–Stokes fluid

Kozono, Hideo; Miura, Masanari; Sugiyama, Yoshie

doi:10.1016/j.jfa.2015.10.016

Cited by 108 publications

(58 citation statements)

References 14 publications

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“…In contrast to (1), in the classical Keller-Segel system the chemoattractant is produced by the bacteria themselves and not consumed (accounting for terms +n−c in place of −nc in the second equation of (1)), and models of Keller-Segel-Stokes type have also been considered ( [40,3]). In Ω = R 3 , mild solutions to a system encompassing both mechanisms at the same time were proven to exist under a smallness condition on initial data ( [20]). Chemotaxis fluid models including logistic growth (κ, µ > 0) have been treated in [38,33,36,43,4].…”

Section: Introductionmentioning

confidence: 99%

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

Lankeit

2016

Math. Models Methods Appl. Sci.

165

131

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Long-term behaviour in a chemotaxis-fluid system with logistic source Johannes LankeitWe consider the coupled chemotaxis Navier-Stokes model with logistic source termsin a bounded, smooth domain Ω ⊂ R 3 under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u and with given functions f ∈ L ∞ (Ω × (0, ∞)) satisfying certain decay conditions and Φ ∈ C 1+β (Ω) for some β ∈ (0, 1). We construct weak solutions and prove that after some waiting time they become smooth and finally converge to the semi-trivial steady state ( κ µ , 0, 0).

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Section: Introductionmentioning

confidence: 99%

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

Lankeit

2016

Math. Models Methods Appl. Sci.

165

131

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“…for all s ∈ (s 0 , T max ). Thus, combination of (13) with (15) and (16) yields that there is a constant…”

Section: Boundedness: Proof Of Theorem 11mentioning

confidence: 99%

Global existence and asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

Cao

Kurima

Mizukami

2018

Math Methods in App Sciences

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This paper considers the 2‐species chemotaxis‐Stokes system with competitive kinetics false(n1false)t+u·∇n1=normalΔn1−χ1∇·false(n1∇cfalse)+μ1n1false(1−n1−a1n2false),x∈normalΩ,1emt>0,false(n2false)t+u·∇n2=normalΔn2−χ2∇·false(n2∇cfalse)+μ2n2false(1−a2n1−n2false),x∈normalΩ,1emt>0,ct+u·∇c=normalΔc−false(αn1+βn2false)c,x∈normalΩ,1emt>0,ut=normalΔu+∇P+false(γn1+δn2false)∇ϕs,1em∇·u=0,x∈normalΩ,1emt>0 under homogeneous Neumann boundary conditions in a 3‐dimensional bounded domain normalΩ⊂R3 with smooth boundary. Both chemotaxis‐fluid systems and 2‐species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2‐species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (u·∇)u in the fluid equation were established in the 2‐dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3‐dimensional case when maxfalse{χ1,χ2false}minfalse{μ1,μ2false}false‖c0‖L∞false(normalΩfalse) is sufficiently small.

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“…Chemotaxis-fluid systems describing a signal being produced by the cells themselves, as with the Keller-Segel type choice g(n, c) = +n − c, up to now have been studied in much fewer works. Global solutions were found to exist in a whole-space setting in the sense of mild solutions [19], in systems with sensitivity functions that obey an estimate of the form χ(n, c) ≤ (1 + n) −α for some α > 0 ( [34,36]), in the presence of additional logistic source terms ( [31]), nonlinear diffusion ( [23]) or for sublinear signal production, that is, g generalizing g(n, c) = n α − c for some α ∈ (0, 1) ([3]).…”

Section: Chemotaxis-fluid Modelsmentioning

confidence: 99%