Special Issue on Modeling and Related Problems of Cross Diffusion PhenomenaNo Access

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

Johannes Lankeit

Johannes Lankeit

Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

E-mail Address: jlankeit@math.uni-paderborn.de

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https://doi.org/10.1142/S021820251640008XCited by:144 (Source: Crossref)

Abstract

We consider the coupled chemotaxis Navier–Stokes model with logistic source terms:

n t + u \cdot \nabla n = Δ n - χ \nabla \cdot (n \nabla c) + κ n - μ n 2, n_{t} + u \cdot \nabla n = Δ n - χ \nabla \cdot (n \nabla c) + κ n - μ n^{2}, <math display="block" altimg="eq-00001.gif"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false" class="MathClass-bin">+</mo><mi>u</mi><mo stretchy="false" class="MathClass-bin">\cdot</mo><mo class="MathClass-op">\nabla</mo><mi>n</mi><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>n</mi><mo stretchy="false" class="MathClass-bin">-</mo><mi>χ</mi><mo class="MathClass-op">\nabla</mo><mo stretchy="false" class="MathClass-bin">\cdot</mo><mo class="MathClass-open" stretchy="false">(</mo><mi>n</mi><mo class="MathClass-op">\nabla</mo><mi>c</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">+</mo><mi>κ</mi><mi>n</mi><mo stretchy="false" class="MathClass-bin">-</mo><mi>μ</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo class="MathClass-punc">,</mo></mrow></math>

c t + u \cdot \nabla c = Δ c - n c, c_{t} + u \cdot \nabla c = Δ c - n c, <math display="block" altimg="eq-00002.gif"><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false" class="MathClass-bin">+</mo><mi>u</mi><mo stretchy="false" class="MathClass-bin">\cdot</mo><mo class="MathClass-op">\nabla</mo><mi>c</mi><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>c</mi><mo stretchy="false" class="MathClass-bin">-</mo><mi>n</mi><mi>c</mi><mo class="MathClass-punc">,</mo></mrow></math>

u t + (u \cdot \nabla) u = Δ u + \nabla P + n \nabla Φ + f, \nabla \cdot u = 0 u_{t} + (u \cdot \nabla) u = Δ u + \nabla P + n \nabla Φ + f, \nabla \cdot u = 0 <math display="block" altimg="eq-00003.gif"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false" class="MathClass-bin">+</mo><mo class="MathClass-open" stretchy="false">(</mo><mi>u</mi><mo stretchy="false" class="MathClass-bin">\cdot</mo><mo class="MathClass-op">\nabla</mo><mo class="MathClass-close" stretchy="false">)</mo><mi>u</mi><mo class="MathClass-rel">=</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo stretchy="false" class="MathClass-bin">+</mo><mo class="MathClass-op">\nabla</mo><mi>P</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>n</mi><mo class="MathClass-op">\nabla</mo><mi mathvariant="normal">Φ</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>f</mi><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mo class="MathClass-op">\nabla</mo><mo stretchy="false" class="MathClass-bin">\cdot</mo><mi>u</mi><mo class="MathClass-rel">=</mo><mn>0</mn></mrow></math>

in a bounded, smooth domain

$Ω \subset ℝ^{3}$ under homogeneous Neumann boundary conditions for

$n$ and

$c$ and homogeneous Dirichlet boundary conditions for

$u$ and with given functions

$f \in L^{\infty} (Ω \times (0, \infty))$ satisfying certain decay conditions and

$Φ \in C^{1 + β} (\bar{Ω})$ for some

$β \in (0, 1)$ . We construct weak solutions and prove that after some waiting time they become smooth and finally converge to the semi-trivial steady state

$(\frac{κ}{μ}, 0, 0)$ .

Communicated by M. Winkler

Keywords:

AMSC: 35B40, 35K55, 35B65, 35Q30, 92C17

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Vol. 26, No. 11

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History

Received 31 January 2016

Revised 9 May 2016

Accepted 6 July 2016

Published: 30 September 2016

Keywords

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System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

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

Abstract

References

Global solvability and eventual smoothness in a chemotaxis-fluid system with weak logistic-type degradation

Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system with subcritical sensitivity

Asymptotic profile of a two-dimensional Chemotaxis–Navier–Stokes system with singular sensitivity and logistic source

Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system

Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion

Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

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

Abstract

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