Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

Jin, Hong-Ying

doi:10.3934/dcds.2018155

Cited by 13 publications

(6 citation statements)

References 42 publications

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“…Whereas the latter, namely, is known to enforce finite-time singularity formation within large classes of two-dimensional and higher dimensional scenarios [18][19][20], no-flux type boundary value problems for (1.3) admit globally defined solutions, either in classical or at least in certain generalized frameworks, under very mild assumptions on the nonlinearity 𝜙 including those from (1.2), and in domains of arbitrary dimension ( [15,[21][22][23][24][25]; cf. also [26][27][28][29][30][31] and [32] for related results in this direction). Apart from that, however, part of the results concerning large time asymptotics indicates that (1.…”

Section: Introductionmentioning

confidence: 85%

“…Proof. By following a standard contraction mapping argument [27], we obtain T max,𝜀 ∈ (0, ∞] as well as functions…”

Section: Preliminaries: Basic Regularity Features Of Approximate Solu...mentioning

confidence: 99%

“…Whereas the latter, namely, is known to enforce finite‐time singularity formation within large classes of two‐dimensional and higher dimensional scenarios [18–20], no‐flux type boundary value problems for () admit globally defined solutions, either in classical or at least in certain generalized frameworks, under very mild assumptions on the nonlinearity

\phi

including those from (), and in domains of arbitrary dimension ([15, 21–25]; cf. also [26–31] and [32] for related results in this direction). Apart from that, however, part of the results concerning large time asymptotics indicates that () yet maintains a significant tendency toward support of spatial structures in the flavor of infinite‐time singular aggregation [24, 33, 34], or also of nonconstant elements contained in corresponding steady state sets [21].…”

Section: Introductionmentioning

confidence: 97%

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Keller–Segel–Stokes interaction involving signal‐dependent motilities

Tian

Winkler

2023

Math Methods in App Sciences

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The chemotaxis‐Stokes system is considered along with homogeneous boundary conditions of no‐flux type for and , and of Dirichlet type for , in a smoothly bounded domain . Under the assumption that , that is bounded on each of the intervals with arbitrary , and that with some and , we have It is shown that for any suitably regular initial data, an associated initial‐boundary value problem admits a global very weak solution.

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Section: Introductionmentioning

confidence: 85%

“…Proof. By following a standard contraction mapping argument [27], we obtain T max,𝜀 ∈ (0, ∞] as well as functions…”

Section: Preliminaries: Basic Regularity Features Of Approximate Solu...mentioning

confidence: 99%

\phi

Section: Introductionmentioning

confidence: 97%

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Keller–Segel–Stokes interaction involving signal‐dependent motilities

Tian

Winkler

2023

Math Methods in App Sciences

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“…For more references about the chemotaxis-uid system, the corresponding global solvability of classical solutions has been investigated by [4][5][6][7][8][9][10][11][12] in two or three-dimensional situations. We also mention complicated variants, e.g., involving rotational ux [13][14][15][16][17][18][19][20] and logistic source terms [21][22][23][24][25][26] as well as nonlinear di usion [4,9,16,[27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Blow-Up Solution in Logarithmic Sensitivity and Indirect Signal Generation Keller–Segel System

Xie

2022

Mathematical Problems in Engineering

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In this paper, we consider the following indirect signal generation and logarithmic sensitivity ε n t = Δ n − χ ∇ · n ∇ ln c x ∈ Ω , t > 0 c t = Δ c − c + w x ∈ Ω , t > 0 w t = Δ w − w + n x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a ball domain Ω ⊂ R N N ≥ 4 with smooth boundary ∂ Ω . This paper considers in the singular limit ε ⟶ 0 ; the result comes from the finite time blow-up of arbitrary large values of n in the corresponding nonlocal scalar parabolic equation case when N ≥ 4 and χ > 2 N / N − 2 .

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“…For corresponding quadratic degradation, in the two-dimensional setting, the existence of global weak solutions was established by the authors in [7] with ρ = 0, or global bounded classical solutions were constructed in [14,30] for the corresponding Navier-Stokes system when ρ ≥ 0 and r > 0. In the three-dimensional setting, the authors in [29] obtained the existence of global bounded classical solutions whenever r > 23.…”

mentioning

confidence: 99%

Approaching constant steady states in a Keller-Segel-Stokes system with subquadratic logistic growth

Tian

2022

DCDS-B

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In this paper, we investigate the large time behavior of the generalized solution to the Keller-Segel-Stokes system with logistic growth <inline-formula><tex-math id="M1">\begin{document}$ \rho n-rn^{\alpha } $\end{document}</tex-math></inline-formula> in a bounded domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb R^d $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M3">\begin{document}$ (d\in\{2, 3\}) $\end{document}</tex-math></inline-formula>, as given by<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} &n_t+{{\bf{u}}}\cdot\nabla n = \Delta n-\chi\nabla\cdot\big(n\nabla c\big)+\rho n-rn^{\alpha }, \\ &c_t+{{\bf{u}}}\cdot\nabla c = \Delta c-c+n, \\ &{{\bf{u}}}_t+\nabla P = \Delta{{\bf{u}}}+n\nabla\phi, \\ &\nabla\cdot{{\bf{u}}} = 0 \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula>for the unknown <inline-formula><tex-math id="M4">\begin{document}$ (n, c, {{\bf{u}}}, P) $\end{document}</tex-math></inline-formula>, with prescribed and suitably smooth <inline-formula><tex-math id="M5">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. Our result shows that if <inline-formula><tex-math id="M6">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \rho $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ r $\end{document}</tex-math></inline-formula> satisfy<disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha > \frac{2d-2}{d}\quad\mathrm{and}\quad\chi^2< K\rho^{ \frac{\alpha -3}{\alpha -1}}r^{ \frac{2}{\alpha -1}} $\end{document} </tex-math></disp-formula>with some positive constant <inline-formula><tex-math id="M10">\begin{document}$ K $\end{document}</tex-math></inline-formula> depending on <inline-formula><tex-math id="M11">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>, the generalized solution converges to a constant steady state ((<inline-formula><tex-math id="M14">\begin{document}$ \frac{\rho}{r})^{ \frac{1}{\alpha -1}}, ( \frac{\rho}{r})^{ \frac{1}{\alpha -1}}, {\bf 0} $\end{document}</tex-math></inline-formula>) after a large time. Our proof is based on the decay property of a functional involving <inline-formula><tex-math id="M15">\begin{document}$ n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M16">\begin{document}$ c $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ {\bf{u}} $\end{document}</tex-math></inline-formula>.

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Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

Abstract: This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity       16 with K 0 = max 0≤c≤∞ |χ(c)| 2 d(c) .

Cited by 13 publications

References 42 publications

Keller–Segel–Stokes interaction involving signal‐dependent motilities

Keller–Segel–Stokes interaction involving signal‐dependent motilities

Blow-Up Solution in Logarithmic Sensitivity and Indirect Signal Generation Keller–Segel System

Approaching constant steady states in a Keller-Segel-Stokes system with subquadratic logistic growth

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