Global Boundedness and Stabilization in a Two-Competing-Species Chemotaxis-Fluid System with Two Chemicals

Zheng, Pan; Willie, Robert; Mu, Chunlai

doi:10.1007/s10884-019-09797-4

Cited by 15 publications

(4 citation statements)

References 32 publications

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“…Remark In three‐dimensional case, we derive global existence of weak solutions to system , which become eventually smooth after some waiting time. Moreover, by means of the arguments in Zheng et al, the asymptotic properties can be also obtained in this situation.…”

Section: Introductionmentioning

confidence: 92%

“…As we mentioned above, both chemotaxis‐Navier‐Stokes systems and two‐competing‐species with two signals are studied by some mathematicians. Recently, the uniform boundedness and stabilization of classical solutions to system under the two‐dimensional case have been derived in Zheng et al However, the above mixed system in three‐dimensional seems to be not investigated yet. The main purpose of this paper is to study the global existence and eventual smoothness of weak solutions to system .…”

Section: Introductionmentioning

confidence: 99%

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Global weak solutions and eventual smoothness in a 3D two‐competing‐species chemotaxis‐Navier‐Stokes system with two consumed signals

Zheng

Willie

2020

Math Methods in App Sciences

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This paper deals with a two‐competition‐species chemotaxis‐Navier‐Stokes system with two different consumed signals false(n1false)t+boldu·∇n1=d1normalΔn1−χ1∇·false(n1∇cfalse)+μ1n1false(1−n1−a1n2false),in3.0235ptnormalΩ×false(0,∞false),ct+boldu·∇c=d2normalΔc−α1cn2,in3.0235ptnormalΩ×false(0,∞false),false(n2false)t+boldu·∇n2=d3normalΔn2−χ2∇·false(n2∇vfalse)+μ2n2false(1−a2n1−n2false),in3.0235ptnormalΩ×false(0,∞false),vt+boldu·∇v=d4normalΔv−α2vn1,in3.0235ptnormalΩ×false(0,∞false),boldut+false(boldu·∇false)boldu=normalΔboldu+∇P+false(β1n1+β2n2false)∇ϕ,in3.0235ptnormalΩ×false(0,∞false),∇·boldu=0,in3.0235ptnormalΩ×false(0,∞false), in a smooth bounded domain normalΩ⊂double-struckR3 under zero Neumann boundary conditions for n1,n2,c,v, and homogeneous Dirichlet boundary condition for boldu, where the parameters di ( i=1,2,3,4) and χj,μj,aj,αj,βj ( j=1,2) are positive. This system describes the evolution of two‐competing species which react on two different chemical signals in a liquid surrounding environment. Recently, the boundedness and stabilization of classical solutions to the above system under two‐dimensional case have been derived in the previous works. However, to the best of our knowledge, the well‐posedness problem of solutions for the above system is still open in the three dimensional setting, because of the difficulties in the Navier‐Stokes system. The aim of this paper is to construct global weak solutions and show that after some waiting time, these weak solutions become eventually smooth.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Global weak solutions and eventual smoothness in a 3D two‐competing‐species chemotaxis‐Navier‐Stokes system with two consumed signals

Zheng

Willie

2020

Math Methods in App Sciences

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“…Nowadays, many authors have also proposed the variants of (2) and obtained lots of interesting results [29,30,32]. It is worth mentioning that in theoretical immunology, the researches on the immune system with chemotaxis have attracted many biologists and math-ematicians [6-9, 11, 21, 24].…”

Section: Introductionmentioning

confidence: 99%

Global boundedness and asymptotic behavior of solutions to a three-dimensional immune chemotaxis system

Gao,

Miao

2024

Z. Angew. Math. Phys.

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In a bounded smooth domain Ω ⊂ R 3 and with a positive parameter χ > 0, this paper is devoted to investigate the global boundedness and asymptotic behavior of solutions to a immune-chemotaxis system. By establishing the uniform boundedness of L ∞ -norm for immune cells and the uniform boundedness of L q -norm with q ∈ (3, ∞) for the chemokines gradient, it is possible to identify a smallness condition on chemosensitivity χ that ensures global existence of classical solution to the model in a three-dimensional bounded domain.Moreover, building on this boundedness property, it is proved that if χ is appropriately small, as time tends to infinity, all the solution will decay exponentially to the spatially constant nontrivial equilibria in an appropriate sense.

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“…Recently, when −(α 1 n 1 + α 2 n 2 )c is replaced with −c+α 1 n 1 +α 2 n 2 in (3), Cao et al [6] studied the global boundedness and stabilization of classical solutions to (3) in three-dimensional case provided that µ 1 and µ 2 are sufficiently large. Moreover, Zheng et al [38] considered the global asymptotic stability for two-species chemotaxis-competition-fluid system with two signals.…”

mentioning

confidence: 99%

Global Boundedness and Stabilization in a Two-Competing-Species Chemotaxis-Fluid System with Two Chemicals

Cited by 15 publications

References 32 publications

Global weak solutions and eventual smoothness in a 3D two‐competing‐species chemotaxis‐Navier‐Stokes system with two consumed signals

Global weak solutions and eventual smoothness in a 3D two‐competing‐species chemotaxis‐Navier‐Stokes system with two consumed signals

Global boundedness and asymptotic behavior of solutions to a three-dimensional immune chemotaxis system

Asymptotic stability in a chemotaxis-competition system with indirect signal production

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