Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: The 3D case

Wang, Yulan; Xiang, Zhaoyin

doi:10.1016/j.jde.2016.07.010

Cited by 118 publications

(46 citation statements)

References 46 publications

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“…Indeed, in the case of the chemotaxis-fluid system without including the chemotactic force on fluid, we can immediately establish the regularity for u once we have the corresponding L p estimate of n (see e.g. [15], [18], [19], [31], [29], [30], [37]). This is not valid in the present setting.…”

Section: Yulan Wangmentioning

confidence: 99%

Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system

Wang¹

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper we deal with the initial-boundary value problem for chemotaxis-fluid model involving more complicated nonlinear coupling term, precisely, the following self-consistent system        nt + u • ∇n = ∆n m − ∇ • n∇c + ∇ • n∇φ , (x, t) ∈ Ω × (0, T), ct + u • ∇c = ∆c − nc, (x, t) ∈ Ω × (0, T), ut + (u • ∇)u + ∇P = ∆u − n∇φ + n∇c, (x, t) ∈ Ω × (0, T), ∇ • u = 0, (x, t) ∈ Ω × (0, T), where Ω ⊂ R 2 is a bounded domain with smooth boundary. The novelty here is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid is considered, which leads to the stronger coupling than usual chemotaxis-fluid model studied in the most existing literatures. To the best of our knowledge, there is no global solvability result on this chemotaxis-Navier-Stokes system in the past works. It is proved in this paper that global weak solutions exist whenever m > 1 and the initial data is suitably regular. This extends a result by Di Francesco, Lorz and Markowich (Discrete Cont. Dyn. Syst. A 28 (2010)) which asserts global existence of weak solutions under the constraint m ∈ (3 2 , 2] in the corresponding Stokes-type simplified system.

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Section: Yulan Wangmentioning

confidence: 99%

Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system

Wang¹

2020

Discrete &Amp; Continuous Dynamical Systems - S

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“…Despite these challenges, some comprehensive results on the global-boundedness and large time behavior of solutions are available in the literature (see [3,19,20,25,27,29,33,34,35] for example). It has been shown that when S = S(x, ρ, c) is a tensor fulfilling |S(x, ρ, c)| ≤ C S (1 + ρ) α for some α > 0 and C S > 0, (1.4) the three-dimensional system (1.3) with µ = 0, κ = 0 admits globally bounded weak solutions for α > 1/2 [27], which is slightly stronger than the corresponding subcritical assumption α > 1/3 for the fluid-free system. As for α ≥ 0, when the suitably regular initial data (ρ 0 , c 0 , u 0 ) fulfill a smallness condition, (1.3) with µ = 0, κ = 1 possesses a global classical solution which decays to (ρ 0 ,ρ 0 , 0) exponentially withρ 0 = 1 |Ω| Ω ρ 0 (x)dx [38].…”

Section: Introductionmentioning

confidence: 99%

Global boundedness and decay property of a three-dimensional Keller–Segel–Stokes system modeling coral fertilization

Pang

Wang

2019

Nonlinearity

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This paper is concerned with the four-component Keller-Segel-Stokes system modelling the fertilization process of corals:subject to the boundary conditions ∇c · ν = ∇m · ν = (∇ρ − ρS(x, ρ, c)∇c) · ν = 0 and u = 0, and suitably regular initial data⊂ R 3 is a bounded domain with smooth boundary ∂Ω. This system describes the spatiotemporal dynamics of the population densities of sperm ρ and egg m under a chemotactic 1 process facilitated by a chemical signal released by the egg with concentration c in a fluidflow environment u modeled by the incompressible Stokes equation. In this model, the chemotactic sensitivity tensor S ∈ C 2 (Ω × [0, ∞) 2 ) 3×3 satisfies |S(x, ρ, c)| ≤ C S (1 + ρ) −αwith some C S > 0 and α ≥ 0. We will show that for α ≥ 1 3 , the solutions to the system are globally bounded and decay to a spatially homogeneous equilibrium exponentially as time goes to infinity. In addition, we will also show that, for any α ≥ 0, a similar result is valid when the initial data satisfy a certain smallness condition.

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“…Indeed, if in (1.2) the summand −∇·(n∇c) is replaced by −∇ · (nS(n)∇c) with S suitably generalizing the prototype given by S(s) = (s + 1) −α for all s ≥ 0 and some α > 0, then known results assert global existence of bounded solutions to a corresponding initial-boundary value problem when in the context of (1.6) we have m + α > 7 6 ([36]), which in the particular case α = 0 considered here rediscovers (1.4) and is thereby stronger than (1.10). An interesting open problem, partially addressed in [33], [34] and [35], consists in determining optimal conditions on the interplay between these two mechanisms which indeed prevent explosions.…”

Section: Resultsmentioning

confidence: 99%

Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

Winkler

2018

Journal of Differential Equations

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A class of chemotaxis-Stokes systems generalizing the prototypeis considered in bounded convex three-dimensional domains, where φ ∈ W 2,∞ (Ω) is given.The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that m > 9 8 .(0.1) Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state ( 1 |Ω| Ω n 0 , 0, 0) in the large time limit. This extends previous results which either relied on different and apparently less significant energytype structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1).

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Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: The 3D case

Cited by 118 publications

References 46 publications

Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system

Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system

Global boundedness and decay property of a three-dimensional Keller–Segel–Stokes system modeling coral fertilization

Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

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