Global existence and boundedness of chemotaxis-fluid equations to the coupled Solow-Swan model

Jie Wu; Zheng Yang; Jie Wu; Zheng Yang

doi:10.3934/math.2023912

AIMS Mathematics

2023, Volume 8, Issue 8: 17914-17942. doi: 10.3934/math.2023912

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Research article Special Issues

Global existence and boundedness of chemotaxis-fluid equations to the coupled Solow-Swan model

Jie Wu ^{1
,
,},
Zheng Yang ²

1.
College of Computer Science, Chengdu University, Chengdu 610106, China
2.
Sichuan University-Pittsburgh Institute, Chengdu 610207, China

Received: 02 February 2023 Revised: 17 May 2023 Accepted: 17 May 2023 Published: 24 May 2023
MSC : 35B65, 35Q35, 35Q92, 92C17

In this paper, we consider the following Keller-Segel-(Navier)-Stokes system to the coupled Solow-Swan model

$ \begin{equation*} \left\{ \begin{split} &n_t+u\cdot\nabla n = \Delta{n}-\chi\nabla\cdot\big(n\nabla{c}\big)+\mu_1 n-\mu_2n^k, \quad &x\in\Omega, \, t>0, \\ &c_t+u\cdot\nabla c = \Delta{c}-c+\mu_3c^\alpha w^{1-\alpha}, \quad &x\in\Omega, \, t>0, \\ &w_t+u\cdot\nabla w = \Delta w-w+n, \quad &x\in\Omega, \, t>0, \\ &u_t+\kappa(u\cdot\nabla u) = \Delta u-\nabla P+n\nabla\Phi, \quad\nabla\cdot u = 0, &x\in\Omega, \, t>0, \end{split} \right. \end{equation*} $

in a smooth bounded domain $ \Omega\subset\mathbb{R}^N\, \, (N = 2, 3) $ with no-flux boundary for $ n, c, w $ and no-slip boundary for $ u $, where the parameters $ \chi > 0, \, \alpha\in(0, 1), \, \mu_1\in\mathbb{R}, \, \mu_2\geq0, \, \mu_3 > 0 $ and $ \kappa\in\{0, \, 1\}, k\geq{N} $. Due to the interference of the fractional nonlinear term of the Solow-Swan model, we use the Moser-Trudinger inequality to obtain the global existence of the solution for two-dimensional case without logistic source. For three-dimensional case, we control the required estimation with the help of the negative term of logistic source to obtain the boundedness and asymptotic behavior. In the process of estimating the corresponding term, we find the order of the negative term of the logistic source is related to the spatial dimension, and we give the decay estimate of the corresponding solutions when $ \mu_1 < 0 $ or $ \mu_1 = 0, \, \mu_2 > 0 $.
- Keller-Segel-Solow-Swan,
- indirect signal production,
- global existence and boundedness,
- asymptotic behavior
Citation: Jie Wu, Zheng Yang. Global existence and boundedness of chemotaxis-fluid equations to the coupled Solow-Swan model[J]. AIMS Mathematics, 2023, 8(8): 17914-17942. doi: 10.3934/math.2023912

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Abstract

In this paper, we consider the following Keller-Segel-(Navier)-Stokes system to the coupled Solow-Swan model

$ \begin{equation*} \left\{ \begin{split} &n_t+u\cdot\nabla n = \Delta{n}-\chi\nabla\cdot\big(n\nabla{c}\big)+\mu_1 n-\mu_2n^k, \quad &x\in\Omega, \, t>0, \\ &c_t+u\cdot\nabla c = \Delta{c}-c+\mu_3c^\alpha w^{1-\alpha}, \quad &x\in\Omega, \, t>0, \\ &w_t+u\cdot\nabla w = \Delta w-w+n, \quad &x\in\Omega, \, t>0, \\ &u_t+\kappa(u\cdot\nabla u) = \Delta u-\nabla P+n\nabla\Phi, \quad\nabla\cdot u = 0, &x\in\Omega, \, t>0, \end{split} \right. \end{equation*} $

in a smooth bounded domain $ \Omega\subset\mathbb{R}^N\, \, (N = 2, 3) $ with no-flux boundary for $ n, c, w $ and no-slip boundary for $ u $, where the parameters $ \chi > 0, \, \alpha\in(0, 1), \, \mu_1\in\mathbb{R}, \, \mu_2\geq0, \, \mu_3 > 0 $ and $ \kappa\in\{0, \, 1\}, k\geq{N} $. Due to the interference of the fractional nonlinear term of the Solow-Swan model, we use the Moser-Trudinger inequality to obtain the global existence of the solution for two-dimensional case without logistic source. For three-dimensional case, we control the required estimation with the help of the negative term of logistic source to obtain the boundedness and asymptotic behavior. In the process of estimating the corresponding term, we find the order of the negative term of the logistic source is related to the spatial dimension, and we give the decay estimate of the corresponding solutions when $ \mu_1 < 0 $ or $ \mu_1 = 0, \, \mu_2 > 0 $.

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